Continuous transmit focusing method and apparatus for ultrasound imaging system

ABSTRACT

In one embodiment, an ultrasound imaging method comprises: providing a probe that includes one or more transducer elements for transmitting and receiving ultrasound waves; generating a sequence of spatially distinct transmit beams which differ in one or more of origin and angle; determining a transmit beam spacing substantially based upon a combination of actual and desired transmit beam characteristics, thereby achieving a faster echo acquisition rate compared to a transmit beam spacing based upon round-trip transmit-receive beam sampling requirements; storing coherent receive echo data, from two or more transmit beams of the spatially distinct transmit beams; combining coherent receive echo data from at least two or more transmit beams to achieve a substantially spatially invariant synthesized transmit focus at each echo location; and combining coherent receive echo data from each transmit firing to achieve dynamic receive focusing at each echo location.

CROSS-REFERENCES TO RELATED APPLICATIONS

This application is based on provisional patent application no.60/701,812, filed Jul. 22, 2005, and entire disclosure of which isincorporated herein by reference.

BACKGROUND OF THE INVENTION

This invention relates generally to ultrasound imaging systems and, moreparticularly, to continuous transmit focusing for ultrasound imagingsystems.

Conventional ultrasound imaging systems generally form an image in thefollowing manner. A short acoustic pulse is transmitted into a region ofinterest from a subset of transducer elements on an array, focused at aparticular depth and direction. The acoustic wavefront formed by thesuperposition of the transmitted pulses propagates along the selecteddirection and upon backscattering from structures contained within theregion of interest, propagates back towards the transducer array (referto FIG. 1). These echos received from different transducer elements aresubsequently amplified and combined using delay, phase, and apodizationin such a manner as to provide a dynamic receive focus which changes asa function of time/depth along the transmitted wavefront direction. Thecombined signal is then log detected and further processed prior tobeing stored. This process is repeated many times as the transmit andreceive directions are changed in such a way as to sweep through theregion of interest, i.e., steered, translated, or both. Upon collectingthe desired number of line acquisitions, this acoustic data is then scanconverted for display to form the resulting ultrasound image. The rateat which these images are formed and displayed is referred to as theframe rate.

For an ultrasound imaging system to produce high quality images, theregion of interest must be properly sampled acoustically, both in therange and azimuth (lateral) dimensions in order to prevent aliasingartifacts, which can arise in all sampled systems. In the rangedimension, the Nyquist sampling theorem requires that an adequate numberof samples in range be acquired based upon the combined round-triptransmit/receive pulse bandwidth. In the azimuth, or lateral, dimension,the Nyquist sampling theorem requires that 1) the region of interest belaterally insonified by a sufficient number of transmit beams and 2) anadequate number of combined round-trip transmit/receive beams laterallysample the region of interest. Stated another way, the Nyquist samplingtheorem imposes a transmit acquisition lateral sampling criteria, aswell as a round-trip transmit/receive lateral sampling criteria. TheNyquist transmit beam spacing Δx_(xmt) is dependent upon the transmitaperture size A_(xmt), focusing location r_(xmt,) and the carrierwavelength of the acoustic radiation λ₀. It is given byΔx _(xmt)=λ₀ F _(xmt)  (1)where the transmit F-number F_(xmt)=r_(xmt)/A_(xmt). The Nyquistround-trip beam spacing Δx is dependent upon both transmit and receiveF-numbers F_(xmt) and F_(rcv) respectively. It is given by

$\begin{matrix}{{\Delta\; x} = {{\lambda_{0}\frac{F_{xmt}F_{rcv}}{\sqrt{F_{xmt}^{2} + F_{rcv}^{2}}}} = \frac{\Delta\; x_{xmt}\Delta\; x_{rcv}}{\sqrt{{\Delta\; x_{xmt}^{2}} + {\Delta\; x_{rcv}^{2}}}}}} & (2)\end{matrix}$where F_(rcv)=r_(rcv)/A_(rcv) for focusing location r_(rcv), andΔx_(rcv)=λ₀F_(rcv). Note that when the receive F-number is much lowerthan the transmit F-number, the Nyquist round-trip beam spacing Δx isdominated by the receive beam characteristics. This is the direct resultof the round-trip transmit/receive beampattern S(ω,x,r) being equal tothe multiplication of the individual transmit and receive beampatternsS_(xmt)(ω,x,r,r_(xmt)) and S_(rcv)(ω,x,r) respectively, at a particularfrequency f and range r, and is given by

$\begin{matrix}{{s\left( {t,x,r} \right)} = {{\frac{1}{2\pi}{\int{{S\left( {\omega,x,r} \right)}{\mathbb{e}}^{{j\omega}\; t}{\mathbb{d}\omega}\mspace{14mu}{S\left( {\omega,x,r} \right)}}}} = {{{S_{xmt}\left( {\omega,x,r,r_{xmt}} \right)}{S_{rcv}\left( {\omega,x,r} \right)}\mspace{14mu}{s_{xmt}\left( {t,x,r,r_{xmt}} \right)}} = {{\frac{1}{2\;\pi}{\int{{S_{xmt}\left( {\omega,x,r,r_{xmt}} \right)}{\mathbb{e}}^{{j\omega}\; t}{\mathbb{d}\omega}\mspace{14mu}{s_{rcv}\left( {t,x,r} \right)}}}} = {\frac{1}{2\pi}{\int{{S_{rcv}\left( {\omega,x,r} \right)}{\mathbb{e}}^{{j\omega}\; t}{\mathbb{d}\omega}}}}}}}} & (3)\end{matrix}$where x is the lateral spatial coordinate, t is the time coordinate,ω=2πf, and the transmit, receive, and round-trip point spread functions(PSF) are the inverse Fourier transforms of their respectivebeampatterns. If the receive beam is much narrower than the transmitbeam, then the receive beam dominates the round-trip beampattern. FIG. 2depicts the situation where the receive beam F-number is substantiallylower than the transmit F-number and as such, the receive beam, which istypically dynamically focused, i.e., the aperture size and focal pointare increased as a function of time/range so as to maintain constantlateral resolution Δx_(rcv), dominates the round-trip beam pattern. Thetransmit beam is generally focused at a user specified depth. Note thereduction in the round-trip sidelobe clutter energy outside of theround-trip beampattern's mainlobe at the transmit focus location, ascompared to away from the transmit focus location. This is due to themultiplicative influence of the transmit beam on the receive beam. Thetransmit beam can have a more significant influence on the receive beamif lower transmit F-numbers are employed. However, since the transmitbeam is not dynamically focused for all depths, the round-tripbeampattern will display much better lateral resolution at the transmitfocus location compared to away from the transmit focus location. Forexample, if F_(xmt)=F_(rcv), Δx=Δx_(rcv)/√2 at the transmit focuslocation, and Δx=Δx_(rcv) away from the transmit focus location—a 41%degradation in lateral resolution, as well as increased sidelobeclutter. This leads to lateral image non-uniformity, which isundesirable for high quality ultrasound imaging.

Conventional ultrasound imaging systems which form a single receive beamfor each transmit beam as depicted in FIG. 2, necessarily couple theNyquist transmit beam spacing Δx_(xmt) to the Nyquist round-trip beamspacing Δx. Instead of having to fire a minimum of L/Δx_(xmt) transmitbeams to adequately insonify the region of interest, the single beamconventional system fires L/Δx transmit beams to cover the region ofinterest, where L is the lateral extent of the region of interest to beimaged. Using (1) and (2), the potential acoustic acquisition rate isreduced by the factor

$\begin{matrix}{\eta = {\sqrt{1 + \frac{F_{xmt}^{2}}{F_{rcv}^{2}}} = \sqrt{1 + \frac{\Delta\; x_{xmt}^{2}}{\Delta\; x_{rcv}^{2}}}}} & (4)\end{matrix}$Using a typical single focus example of F_(xmt)=2.5 and F_(rcv)=1.0, thereduction in the potential acoustic acquisition rate using (4) is ˜2.7.Almost three times as many transmit beams are being fired than requiredby Nyquist sampling requirements.

If the transmit and receive F-numbers are made equal, i.e.,F_(xmt)=F_(rcv)=1.0, then the reduction in potential frame rate using(4) is ˜1.4, which is not as dramatic as when the receive F-number ismuch lower than the transmit F-number; however, an additional problem isintroduced, namely the limited depth of field of the transmit beam.While the receive beam is dynamically focused, the transmit beam isfocused only at a single depth. The range r over which the transmit beamcan be considered “in focus” is given by the depth of field expression

$\begin{matrix}{{R_{DOF}^{xmt} \sim {{\beta\lambda}\; F_{xmt}^{2}}},{4 \leq \beta \leq {{8\mspace{14mu} r_{xmt}} - \frac{R_{DOF}^{xmt}}{2}}\underset{\_}{<}r\underset{\_}{<}{r_{xmt} + \frac{R_{DOF}^{xmt}}{2}}}} & (5)\end{matrix}$where the choice of β depends upon what phase error is assumed at theend elements of the transmit aperture in the depth of field (DOF)derivation. Note the DOF dependence on the square of the transmitF-number. As the transmit F-number decreases, the transmit beam'slateral resolution increases linearly as given by (1); however, therange over which the transmit beam will be in focus, and have influenceon the round-trip lateral resolution given by (2), decreasesquadratically. This introduces image lateral non-uniformity and higherclutter away from the transmit focus and is undesirable in high qualityultrasound images. The conventional approach to mitigate this behavioris to transmit multiple times along each transmit beam, changing thetransmit focus location r_(xmt) on each firing and performing aconventional receive beam formation on each. The resulting multipleround-trip signals are then typically combined following the detectionprocess, with the DOF image region around each transmit focus r_(xmt)being retained, with the rest discarded. Thus, each detected line is thecomposite of multiple lines, each having a different transmit focus. Thenumber of transmit firings along the same transmit beam direction thatare required is dependent upon the desired transmit F-number to besupported, and the display range of the region of interest, with thenumber of transmit firings going up as the square of the transmitF-number reduction. This produces a significant loss in acousticacquisition rate due to the multiple transmit firings along the sameline.

BRIEF SUMMARY OF THE INVENTION

The reduction in potential acoustic acquisition rate, and ultimatelyframe rate, due to lateral sampling requirements, increased transmitresolution, and reduced round-trip sidelobe clutter, has led to manytechniques being developed which seek to provide a true dynamic transmitfocus and provide a better performance/frame rate tradeoff. Thetechniques developed generally fall into several classes, namely,

-   I. Composite,-   II. Multiple beams,-   III. Transmit sub-apertures,-   IV. Element dependent transmit waveform generation/receive    filtering, and-   V. Deconvolution

I. Composite techniques seek to approximate a dynamic focused transmitbeam by compositing multiple transmit beams of different foci togetheralong the same direction, either fired sequentially in time as discussedabove, or simultaneous in time through linear superposition. If firedsequentially in time, the compositing of transmit beams of differentfoci can be performed following detection by simply discarding receivedata beyond the transmit depth of field and stitching together theremaining detected receive data, which is the conventional sequentialtransmit approach. Or the compositing can be performed usingpre-detected, i.e., coherent, receive data and combined using a weightedsuperposition of the coherent receive data from the multiple transmitfirings. See Synthetic Dynamic Transmit Focus, Brent Robinson & CliffCooley, 2000 IEEE Ultrasonics Symposium, or Synthetic Transmit Focusingfor Ultrasound Imaging, Bruno Haider, 2000 IEEE Ultrasonics Symposium.Compositing the transmit beams of different foci sequentially in time isnot very efficient since progressively less and less information fromeach component transmit firing is being retained, while the number oftransmit firings increases, decreasing the frame rate accordingly.Another technique composites the transmit beams of different focisimultaneous in time by linearly superimposing each transmit channel'sapodized and delayed transmit waveforms together, i.e., compoundfocusing. See U.S. Pat. No. 5,581,517. While the multiple transmitfirings in time and the commensurate decrease in frame rate are avoided,it requires a linear transmitter (which is generally more costly andpower inefficient), round-trip sidelobe clutter is increased around eachfoci compared to a single transmit foci due to the simultaneous transmitbeam's acoustic interference (which limits how close the compoundtransmit foci can be), and the technique does not address the reductionin potential frame rate due to transmit lateral sampling issues. Asimpler, older technique simply used a transmit delay profile which wasa composite of multiple transmit delay profiles each focused atdifferent depths. The outer transmit channel's delay was dominated bythe deeper transmit foci, and the inner transmit channel's delay wasdominated by the shallower transmit foci. However this technique, whilenot reducing the frame rate, improves transmit beamformation performanceaway from the focus, i.e., round-trip sidelobe clutter, at the expenseof beamformation performance at the focus, i.e., poorer resolution, andit still does not address the reduction in potential frame rate due totransmit lateral sampling issues.

II. Multiple beam techniques, either transmit, receive, or both, seek toimprove frame by reducing the number of transmit firings it takes tolaterally sample the region of interest. In the case of multiplesimultaneous transmit beams, along with receive beams aligned along eachtransmit beam, less time is taken to sweep the region of interest toform the image for a given receive line density and thus, higher framerate is achieved. However, the interference between the simultaneoustransmit beams leads to increased acoustic clutter and degrades imagequality. Using multiple receive beams formed from the same transmit beamcan also improve frame rates since less time is required to sweep theregion of interest for a given receive line density. However, receivebeams which are spatially displaced from the transmit beam center canlead to image distortion, more so around the transmit focus, higheracoustic clutter, and results in spatial variance which is undesirablefor high quality imaging. A way to mitigate this distortion wasdescribed in U.S. Pat. No. 6,016,285, using two receive beams from eachtransmit firing and coherently (pre-detection) combining receive beamsfrom two neighboring transmit firings, along with combining receivebeams from the same firing as a way to re-center the receive beams alongtransmit firings. While eliminating geometric distortion, line-to-linegain variations are introduced, which if not removed through spatialfiltering (which reduces lateral resolution) lead to imaging artifacts.A multiple beam approach, Parallel Beamforming using Synthetic TransmitBeams by T. Hergum, T. Bjastad, and H. Torp, 2004 IEEE InternationalUltrasonics, Ferroelectrics, and Frequency Control Joint 50^(th)Anniversary Conference, pages 1401-1404, uses a weighted sum of multiplecoherent receive beams from different transmit beams, forming aninterpolated, or synthesized transmit beam, aligned along each receivebeam. This addresses the lateral spatial sampling component of framerate reduction, however dynamic transmit focusing is not addressed.Therefore, use of multiple beams partially addresses the lateral spatialsampling component of frame rate reduction; however, these classes oftechniques have not addressed the transmit beam focusing aspect and assuch, if increased transmit resolution is desired, multiple beamscombined with some class of compositing is required, with the advantagesand disadvantages described previously.

III. Transmit sub-aperture techniques seek to improve frame rates bytrying to eliminate the process of compositing, or sequential transmitfocusing, altogether by creating a dynamic transmit focus, analogous todynamic receive focusing. The difficulty is that once the transmitwavefront is launched and propagation/diffraction begins, itscharacteristics cannot be directly altered. However, the effectivetransmit focus can be altered after the fact during receive processingby breaking up the transmit aperture into several, or many,sub-apertures (in the limit, each transducer element is firedseparately) where each sub-aperture is fired sequentially in time. SeeMulti-Element Synthetic Transmit Aperture Imaging using TemporalEncoding by K. L. Gammelmark and J. A. Jensen, 2002 SPIE Medical ImagingMeeting, Ultrasonic Imaging and Signal Processing, 2002; pages 1-13. Onreceive, the echoes from each sub-aperture firing can be delayed,phased, apodized, and summed prior to, or after, receive beam formation,varying the delay, phase, and apodization values with range, to form adynamic transmit focus. This has several limitations/complexities—first,it requires linearity in the transmit-receive processing path. Whilethis is true for fundamental imaging, it is not strictly true for 2^(nd)harmonic and higher order non-linear imaging modes. For example, in2^(nd) harmonic imaging, the transmit beam becomes essentially “squared”during the transmit propagation process, effectively reducing sidelobesand clutter in the transmit beam. The subsequent squaring that willoccur during the propagation/diffraction of each transmit sub-apertureand summing on receive will not be equivalent to transmitting the entireoriginal transmit aperture, which is the sum of the transmitsub-apertures, and squaring the result, i.e.,

$\begin{matrix}{{{\sum\limits_{i = 1}^{N_{sub}}\;\left\{ {G\left\lbrack {z_{i}(n)} \right\rbrack} \right\}^{2}} \neq \left\{ {G\left\lbrack {\sum\limits_{i = 1}^{N_{sub}}\;{z_{i}(n)}} \right\rbrack} \right\}^{2}}\therefore{{\sum\limits_{i = 1}^{N_{sub}}\;\left\{ {G\left\lbrack {z_{i}(n)} \right\rbrack} \right\}^{2}} \neq \left\{ {\sum\limits_{i = 1}^{N_{sub}}\;{G\left\lbrack {z_{i}(n)} \right\rbrack}} \right\}^{2}}} & (6)\end{matrix}$where n is the transmit element, N_(sub) is the number of transmitsub-apertures, z_(i)(n) represents the apodized, phased, and delayedtransmit aperture function, and G is the linear propagation/diffractionoperator. In other words, the sum of the squares is not equal to thesquare of the sums. Also, the signal-to-noise ratio (SNR), already aproblem in non-linear imaging modes, will be an even greater problemusing transmit sub-apertures due to the sum of the squares being lessthan the square of the sums. More importantly, to the extent thesetechniques seek to break up the transmit aperture into smaller andsmaller sub-apertures, thus improving the ability to form an effectivedynamic focused transmit beam, the broader and less focused eachsub-aperture's individual transmit beampattern becomes, approaching thatof a point source as the sub-aperture shrinks to a single element, whichwill have a fairly uniform amplitude distribution. Any non-linearimaging mode which reduces transmit beam sidelobe clutter throughsquaring, or higher order non-linearity, will no longer be as effective,i.e., squaring a uniform beam produces a uniform beam. Attempts toimprove the SNR situation by transmitting on multiple elements, delayedand phased in such a way as to create point source wavefront, analogousto a single element, does increase the transmitted signal level;however, the transmit beampattern is still fairly uniform and as such,it will not see much improvement in sidelobe clutter from non-linearpropagation/diffraction. Motion will be an issue in both fundamental andnon-linear imaging modes since to form the entire desired transmitaperture, many firings may be required, i.e., 128, etc., which willrequire a fair amount of time. For example, assuming a carrier frequencyof 3.5 MHz, a depth of 240 mm with an assumed sound speed of 1540 m/s,the round-trip propagation time is 311 μsec, which for 128elements/firings is 40 msec. This is quite a long time for phasecoherency to be maintained across the elements to a fraction of the 3.5MHz wavelength, which is 0.44 mm. Motion during that time will introducetransmit beamformation delay/phase errors due to the many firings ittakes to construct the effective transmit beam, which will degrade thequality of the effective transmit beam characteristics and introduceincreased acoustic clutter and other motion artifacts. Lastly, dependingupon how many transmit sub-apertures are employed, the processingrequirements can be quite high, which can be even higher if motioncompensation techniques are considered, making some of these techniquesimpractical for real-time imaging given the current state of the arttechnology.

IV. Techniques that use element dependent transmit waveform generation,possibly combined with receive filtering, seek to employ differenttransmit waveforms on each element in such a way that during propagationand diffraction, the individual waveforms combine which, when filteredon receive, an effective dynamic focused transmit beam is formed. U.S.Pat. No. 6,108,273 describes starting with a traditional short pulse atthe center of the transmit aperture, whose higher spectral frequencycomponents are progressively advanced in time compared to the lowerfrequency components, as the transmit element location approaches eitherend of the transmit aperture. This produces a chirp-like waveform ofincreasing length away from the transmit aperture center. On receive, abandpass filter is employed whose center frequency is decreased withdepth. The transmitted pulse's higher frequency components that areselectively passed by the depth dependent receive bandpass filter havediffracted from a transmit delay arc of higher curvature, whicheffectively focuses the transmit wavefront at shallower depths. Thetransmitted pulse's lower frequency components that are selectivelypassed by the depth dependent receive bandpass filter have diffractedfrom a transmit delay arc of lower curvature, which effectively focusesthe transmit wavefront at deeper depths. This approximates a dynamicallyfocused transmit beam. The limitations of this approach is that thepotential round-trip pulse bandwidth and hence, axial resolution, iscompromised at the expense of the dynamic transmit focusing effect. Inaddition, this technique alone does not address the lateral spatialsampling component of frame rate reduction, although some of the othertechniques outlined could be used in conjunction with this technique.

V. Techniques that use a deconvolution approach to improve the transmitfocusing characteristics also have various limitations. For example,Retrospective Dynamic Transmit Focusing by S. Freeman, Pai-Chi Li, andM. O'Donnell, Ultrasonic Imaging 17, pages 173-196, (1995), used asingle receive beam system along with a spatial filter of a finitenumber of taps, operating across receive beams. The design of thespatial filter was designed with the intent to deconvolve out theeffects of the defocused transmit beam from the round-trip point spreadfunction. As such, this approach will be limited by the receive beam'sinfluence on the round-trip point spread function, reducing the extentto which the transmit focusing characteristics can be improved. Inaddition, it does not address the lateral spatial sampling component offrame rate reduction.

The present invention seeks to provide high quality imaging using aminimum of transmit firings by decoupling the Nyquist transmit andround-trip spatial sampling requirements, and at the same timemaintaining a continuous transmit focus throughout the region ofinterest, which reduces or eliminates the need for transmit focuscompositing, i.e., sequential transmit focus, both of which dramaticallyimprove the acoustic acquisition rate. Reduction of the number oftransmit firings becomes especially important for ultrasound systemswhich are battery powered in order to conserve energy and reduce powerconsumption. Elimination of transmit focus user controls is also adesirable feature, especially for portable ultrasound systems giventheir compact user interface. The invention works in linear ornon-linear imaging modes, works with standard 1-D and 2-D transducerarrays, can be extended to three spatial dimensions, i.e., 3-D imaging,is compatible with all scan formats, and is robust in presence ofmotion.

An aspect of the present invention is directed to an ultrasound imagingmethod for achieving transmit and receive focusing at every echolocation within a region of interest. The method comprises providing aprobe that includes one or more transducer elements for transmitting andreceiving ultrasound waves; generating a sequence of spatially distincttransmit beams which differ in one or more of origin and angle;determining a transmit beam spacing substantially based upon acombination of actual and desired transmit beam characteristics, therebyachieving a faster echo acquisition rate compared to a transmit beamspacing based upon round-trip transmit-receive beam samplingrequirements; storing coherent receive echo data, from two or moretransmit beams of the spatially distinct transmit beams; combiningcoherent receive echo data from at least two or more transmit beams toachieve a substantially spatially invariant synthesized transmit focusat each echo location, i.e., to achieve transmit synthesis; andcombining coherent receive echo data from each transmit firing toachieve dynamic receive focusing at each echo location.

In some embodiments, the probe is a 1-D, 2-D array with varying degreesof elevation beamforming control, including aperture, delay, and phase,or a general 2-D scanning array, or a sparse 1-D or 2-D array. Thespatially distinct transmit beams are generated electronically,mechanically, or any combination thereof; to scan a 2-D plane or 3-Dvolume. Coherent receive echo data is receive channel data fromdifferent transducer elements. Combining coherent receive echo data fromeach transmit firing to achieve dynamic receive focusing is performedprior to combining coherent receive echo data from at least two or moretransmit beams to achieve transmit synthesis. Combining coherent receiveecho data from each transmit firing to achieve dynamic receive focusingis performed prior to storing coherent receive echo data. Alternatively,combining coherent receive echo data from each transmit firing toachieve dynamic receive focusing is performed subsequent to combiningcoherent receive echo data from at least two or more transmit beams toachieve transmit synthesis. In other embodiments, combining coherentreceive echo data from each transmit firing to achieve dynamic receivefocusing and combining coherent receive echo data from at least two ormore transmit beams to achieve transmit synthesis are performed incombination.

In specific embodiments, combining coherent receive echo data from eachtransmit firing to achieve dynamic receive focusing and combiningcoherent receive echo data from at least two or more transmit beams toachieve transmit synthesis are performed at arbitrary echo locations,i.e., area/volume formation, or along multiple ray-like paths, i.e.,multiple receive beamformation. One or more non-linear components of thetransmitted ultrasound waves are included in the steps of storingcoherent receive echo data, combining coherent receive echo data fromeach transmit firing to achieve dynamic receive focusing, and combiningcoherent receive echo data from at least two or more transmit beams toachieve transmit synthesis. The method may further comprise forming animage with an imaging mode that includes one or more of B, color Doppler(velocity, power), M, spectral Doppler (PW), with or without usingcontrast agents. Velocity imaging modes including color velocity andspectral Doppler (PW) are achieved using motion compensation incombining coherent receive echo data from at least two or more transmitbeams to achieve transmit synthesis. The spectral Doppler (PW) modeinvolves processing one or more sample volumes along a synthesizedtransmit beam obtained by the transmit synthesis.

In some embodiments, combining coherent receive echo data from at leasttwo or more transmit beams to achieve transmit synthesis includescombining using one or more of delay, phase, amplitude, and convolution.Combining coherent receive echo data from at least two or more transmitbeams to achieve transmit synthesis may be responsive to echo location,as well as to the spatial and temporal characteristics of the actualand/or desired transmit beam characteristics, which may include one ormore non-linear components of the transmitted ultrasound waves. Thespatially distinct transmit beams may differ in one or more of delay,phase, apodization, amplitude, frequency, or coding. The spatiallydistinct transmit beams may have a single focus at a predeterminedrange, with an F-number ranging from 0.5 to 10. Two or more of thespatially distinct transmit beams are fired simultaneously, or with atime gap less than a round-trip propagation time, for faster echoacquisition.

Another aspect of the invention is directed to an ultrasound imagingsystem for achieving transmit and receive focusing at every echolocation within a region of interest. The system comprises a probe thatincludes one or more transducer elements for transmitting and receivingultrasound waves; and a processor. The processor generates a sequence ofspatially distinct transmit beams which differ in one or more of originand angle; determines a transmit beam spacing substantially based upon acombination of actual and desired transmit beam characteristics, therebyachieving a faster echo acquisition rate compared to a transmit beamspacing based upon round-trip transmit-receive beam samplingrequirements; stores coherent receive echo data, from two or moretransmit beams of the spatially distinct transmit beams; combinescoherent receive echo data from at least two or more transmit beams toachieve a substantially spatially invariant synthesized transmit focusat each echo location, i.e., to achieve transmit synthesis; and combinescoherent receive echo data from each transmit firing to achieve dynamicreceive focusing at each echo location.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 is a block diagram of the system architecture of a conventionalultrasound imaging system.

FIG. 2 shows transmit and round-trip beam patterns, as well as transmitand round-trip lateral sampling requirements, for conventionalultrasound imaging systems which form a single receive beam for eachtransmit beam.

FIG. 3 is a block diagram of the system architecture of an ultrasoundimaging system according to an embodiment of the present invention.

FIG. 4 shows examples of area formation for a rectilinear scancoordinate system and a curved/steered scan coordinate system.

FIG. 5 shows a comparison between conventional single-beam line-by-lineacoustic acquisition where the transmit and round-trip spatial samplingrequirements are coupled, and zone acquisition of the present inventionwhere the transmit and round-trip spatial sampling requirements aredecoupled.

FIG. 6 illustrates transmit beam sampling for a planar transducer arrayemploying a linear scan format according to an embodiment of the presentinvention.

FIG. 7 shows an architecture topology of image/volume reconstruction(Image/Volume Reconstruction I) according to one embodiment of thepresent invention.

FIG. 8 shows an architecture topology of image/volume reconstruction(Image/Volume Reconstruction II) according to another embodiment of thepresent invention.

FIG. 9 shows plots of transmit, receive, and total computational effortfor transmit sub-apertures and transmit synthesis to compare theircomputation efficiency.

FIG. 10 is a plot of signal-to-noise ratio (SNR) showing improvement asa function of depth for transmit synthesis.

FIGS. 11A-11C show different arrangements of providing a motionprocessor block in the architecture topology of image/volumereconstruction to address motion effects.

DETAILED DESCRIPTION OF THE INVENTION

The present invention seeks to provide high quality imaging using aminimum of transmit firings by decoupling the Nyquist transmit andround-trip spatial sampling requirements, and at the same timemaintaining a continuous transmit focus throughout the region ofinterest, which reduces or eliminates the need for transmit focuscompositing, i.e., sequential transmit focus, both of which dramaticallyimprove the acoustic acquisition rate. Reduction of the number oftransmit firings becomes especially important for ultrasound systemswhich are battery powered in order to conserve energy and reduce powerconsumption. Elimination of transmit focus user controls is also adesirable feature, especially for portable ultrasound systems giventheir compact user interface. The invention works in linear ornon-linear imaging modes, works with standard 1-D and 2-D transducerarrays, can be extended to three spatial dimensions, i.e., 3-D imaging,is compatible with all scan formats, and is robust in presence ofmotion.

FIG. 3 shows a block diagram of the system architecture according to anembodiment of the present invention. However, this is but arepresentative example and many other system architectures can be used.Referring to FIG. 3, following receive front-end TGC, analog-to-digitalconversion and demodulation from RF to baseband is performed. Theresulting in-phase and quadrature, i.e., I/Q, data is then stored into arandom access channel domain memory. The memory can be SDRAM, MRAM, orany other suitable memory device. Equivalently, magnitude and phase datacould be stored. The channel domain memory holds Nt time samples for allNch system channels, for an arbitrary number of transmit firings, i.e.,the number of transmit firings required to insonify the entire region ofinterest—a frame, or more. Next, the channel domain I/Q data is read bythe digital signal processor (DSP) and employing software algorithms,the DSP performs receive reconstruction, detection, and scan conversionfor display. Alternatively, the DSP may consist of a single DSP ormultiple DSP's, or a combination of programmable logic, i.e., FPGA's,and DSP's. The receive reconstruction process is analogous to receivebeamformation; however, it is more general and, in conjunction with therandom access channel domain memory and DSP/software programmablearchitecture, provides arbitrary spatial sampling, allowing it to takeadvantage of spatially dependent sampling requirements and reducecomputation time.

Reconstruction represents the transformation of channel data intopre-detection image data over an area or volume, from the insonificationby a focused, broad, or defocused transmit beam, defined as a transmitzone. The area is defined by a collection of range and azimuthcoordinates, where the azimuth coordinate is defined by a suitablelateral coordinate, i.e., horizontal position x, steering angle θ, etc.This is shown in FIG. 4 which depicts several representative examples.The transducer can be planar, curved, or of some other form, and thescan geometry, i.e., transmit/receive beam origins and angles, can bearbitrary. The random access channel domain I/Q data memory allows theDSP to form an arbitrary I/Q point from any transmit firing, or zone.Alternatively, the DSP can use channel magnitude and phase data to forma magnitude and phase point. The reconstruction software algorithm inthe DSP may include channel delay, phase, apodization, and summationprocessing as is well known in the art to form an image I/Q point, butit is not limited to this particular image formation technique and caninclude non-linear processing elements. In general, the collection ofrange and azimuth coordinates defining a reconstructed I/Q point withinthe area formation region, from any given transmit zone, can bedescribed byr _(ij) =f _(r)(r,x,i,j, . . . ) or r _(ij) =f _(r)(r,θ,i,j, . . . )x _(ij) =f _(x)(r,x,i,j, . . . ) or θ_(ij) =f _(θ)(r,θ,i,j, . . . )  (7)where f_({r,xθ}) describes the functional dependence of the specifiedcoordinate on its input parameters, (i,j) are suitable indexingparameters, and . . . indicates any other parameter dependence for therange and azimuth coordinates, i.e., transmit focus location, etc. Ifthe collection of range and azimuth coordinates defining thereconstructed area are independent and assume a more standard dependenceon the indexing parameters, i.e.,r _(i) =r ₀ +i·Δr i={0,1, . . . N _(r)−1}x _(j) =x ₀ +j·Δx j={0,1, . . . N _(x)−1}orθ_(j)=θ₀ +j·Δθj={0,1, . . . N _(θ)−1}  (8)then area formation reduces to standard receive beamformation, i.e.,ray-like paths. In the case where more than one of these receive beamsare formed simultaneously from a single transmit firing, or zone, thenit is considered multiple receive beamformation, also known in the artas parallel beamformation. In three spatial dimensions, i.e., 3-Dimaging, area formation becomes volume formation and can be described bya collection of suitable range, azimuth, and elevation coordinates. Theexamples shown in FIG. 4 can be easily extended into three spatialdimensions, i.e., if y represents the elevation coordinate, thenr _(ijk) =f _(r)(r,x,y,i,j,k, . . . )x _(ijk) =f _(x)(r,x,y,i,j,k, . . . )y _(ijk) =f _(y)(r,x,y,i,j,k, . . . )  (9)

If angular coordinates are preferred in describing the desired scangeometry, then equation (9) can be suitably modified. If the collectionof range and azimuth coordinates defining the reconstructed volume areindependent and assume a more standard dependence on the indexingparameters, i.e.,r _(i) =r ₀ +i·Δr i={0,1, . . . N _(r)−1}x _(j) =x ₀ +j·Δx j={0,1, . . . N _(x)−1}y _(k) =y ₀ +k·Δy k={0,1, . . . N _(y)−1}  (10)If angular coordinates are preferred in describing the desired scangeometry, then (10) can be suitably modified.

It is important to note that if the transmit and round-trip spatialsampling requirements are decoupled using the system architecture shownin FIG. 3, thereby reducing or eliminating the potential acousticacquisition reduction factor η given by (4), then the frame rate isessentially decoupled from the acoustic acquisition frame rate. Whilethe acoustic acquisition rate is determined by the number of transmitfirings and acoustic propagation time, the frame rate will be determinedby the DSP and software algorithm's computation time required to processthe acquired channel I/Q data and form image I/Q data, i.e., areaformation, perform detection, scan conversion, etc., and display theimage, up until the acoustic acquisition rate becomes the limitingfactor. This characteristic provides the system the opportunity toproduce high quality images at much higher frame rates than conventionalsystems, provided sufficient processing performance is utilized.Alternatively, the system could expend the additional time availableperforming more sophisticated image formation algorithms to furtherimprove image quality, reduce system power consumption by maintainingthe frame rate at conventional system frame rates, etc. The channeldomain data memory serves as a buffer between the acoustic acquisitionrate and the DSP and software algorithm processing rate.

FIG. 5 shows a comparison between conventional single-beam line-by-lineacoustic acquisition where the transmit and round-trip spatial samplingrequirements are coupled, and zone acquisition of the present inventionwhere the transmit and round-trip spatial sampling requirements aredecoupled. Depending upon the transmit F-number, the improvement inacoustic acquisition rate compared to conventional systems can besubstantial as described previously. The process by which the region ofinterest is insonified based upon transmit sampling requirements aloneis termed zone acquisition, from which a relatively large image I/Q areacan be reconstructed from channel I/Q data such that round-trip Nyquistspatial sampling requirements are met or exceeded. To the extent thatarea formation reduces to the special case of multiple receivebeamforming, then the receive beam density must meet or exceed Nyquistround-trip sampling requirements.

The zone acquisition process can be easily extended to three dimensionsto yield even larger improvements, where the large number of transmitfirings typically required by conventional systems results in very lowframe rates. The zone acquisition process has advantages in otherapplications. For example, in contrast agent imaging where micro-bubblesare injected into the body and imaged with ultrasound, either atessentially the same insonification frequency or suitable harmonic,reducing the number of transmit firings and increasing the lateralspacing of the transmit beams results in less bubble destruction whereminimum or no bubble destruction is desired.

The procedure to decouple transmit and round-trip spatial samplingrequirements, as well as provide a continuous transmit focus, is nowdescribed for an embodiment of the present invention. The motivation forthis technique is based upon the following line of reasoning. Aconventional fixed-focused transmit beam produces high quality imagesaround the focus, and works well in harmonic imaging modes, etc.,however lateral resolution and uniformity degrade away from the focus.Rather than reinventing the transmit beam, i.e., decomposing thetransmit aperture into many pieces, only to sum the resulting componentbeam responses back up on receive as outlined in some of the othertechniques, suffering the many disadvantages described, the presentembodiment simply corrects the transmit beam away from the focus asneeded. This can be accomplished using the following insight. Consider asingle receive element on a transducer array receiving the backscatteredsignal from the transmit pulse wavefront impinging on a point scattererin the region of interest. As the transmit beam is swept through theregion of interest, the received backscattered signal from the pointscatterer, on a given receive element, will sample the transmit beamfrom unique spatial locations. In principle, if the transmit beam can beadequately sampled in time as well as in space, then a spatial-temporalfilter can be designed that can transform the actual transmit beam intothat of another transmit beam of differing spatial and temporalcharacteristics, provided that the transmit beam's spatial and temporalcharacteristics are sufficiently predictable. The receive element on thetransducer array provides that sampling. For example, if a pointscatterer in the region of interest lies in the unfocused region of thetransmit beam, i.e., at a depth shallow to the transmit focus, then theevolution of signals received on a given transducer element, as thetransmit beam is swept through the region of interest, will provide arepresentation of the defocused transmit beam.

FIG. 6 shows an example of a planar transducer array employing a linearscan format, where a single point scatterer is located at eitherr=r_(a), (a depth shallow to the transmit focus), r=r_(b) (a depth atthe transmit focus), or r=r_(c) (a depth deeper than the transmitfocus), are shown. The transmit apodization and delay profiles aretranslated across the transducer array so as to translate the transmitbeam across the region of interest. The transmit beam's pulse waveform,after having been subjected to frequency dependent attenuation, 1/rfalloff, the transducer's receive spectral response for the elementshown, as well as any other filter response in the receive signal path,produces the receive waveforms shown. The evolution of these receivewaveforms as the transmit beam is translated across the region ofinterest are shown at times corresponding to a point scatterer depth ofeither r_(a), r_(b), or r_(c). For a given point scatterer depth, thereceive waveform is shifted in time according to how long the transmitbeam's wavefront takes to propagate from the transducer face, to thepoint scatterer, and back to a given receive element, referenced to asuitable time origin, as the transmit beam is translated across theregion of interest. For a point scatterer at r=r_(a), the receivewaveform arrives earlier at the edge of the transmit beam, and later forthe center of the transmit beam due to the focusing curvature. For apoint scatterer at either r=r_(b) or r=r_(c), the receive waveformsdisplay the behavior shown for analogous reasons.

Now consider the demodulated receive waveforms. For a given time/rangesample lying within the evolution of receive waveforms from a givenpoint scatterer depth, plotting the magnitude and phase, orequivalently, real and imaginary, i.e., I/Q, values of the giventime/range sample as the transmit beam is swept through the region ofinterest will result in the beam patterns shown in FIG. 6. At rangelocations r=r_(a) or r=r_(c), away from the transmit focus depth, thebeam pattern is defocused and broad, displaying the classic quadraticphase error. At range location r=r_(b), the beam pattern is focused. Thereal and imaginary beam pattern components display the same behavior.One skilled in the art will recognize the beam patterns away from thetransmit focus as that of a chirp waveform, albeit a spatial chirp asopposed to a temporal chirp, and is the result of the transmit beam'sapproximately parabolic delay profile. Diffraction provides the spatialcompression necessary to transform the defocused transmit beam in thenear field to a focused beam at the focus location, however only at thefocus location. Therefore, in order to produce a focused transmit beamfor all ranges, a spatial filter is required to correct the transmitbeam away from the transmit focus, correcting for diffraction effects,and perform spatial compression throughout the region of interest.

Transmit Synthesis—Spatial Diffraction Transform

The following analysis will consider several examples which are intendedto illustrate the transmit synthesis technique; however, the inventionis not meant to be limited by the examples provided. Note that in theanalysis, continuous time is used for simplicity unless otherwise noted.Sampled time can also be used in an analogous manner.

Consider the architecture topologies shown in FIG. 7 (Image/VolumeReconstruction I) and FIG. 8 (Image/Volume Reconstruction II). Receivedchannel data from each transmit beam firing n is acquired and stored inchannel domain memory. This channel domain data can either be receivedRF, demodulated RF to IF, or demodulated to baseband I/Q signals fromthe transducer elements, or other suitable representation. The sequenceof transmit beam firings can be swept through the region of interest inan arbitrary fashion, either in two or three dimensions. The transmitbeam can be swept through the region of interest either electronically,mechanically, or any combination thereof. The transmit beams fromspatial location to spatial location may differ in characteristics suchas amplitude, phase, delay, carrier frequency, focusing characteristics,waveforms, coding, etc. The transducer employed can be eitherconventional 1-D arrays, 1-1/2-D arrays, or 2-D matrix arrays. In FIG.7, the Transmit Synthesis block precedes the Area/Volume Formationblock, whereas in FIG. 8 the Transmit Synthesis block follows theArea/Volume Formation block. In both cases, the Reconstruction blockcontains both Transmit Synthesis and Area/Volume Formation Blocks. TheTransmit Synthesis block in FIG. 7 consists of transmit beam n, time t,channel ch, and spatial coordinate r _(ijk) dependent filters h^(syn)whose outputs are summed together across N_(syn)(r _(ijk)) transmit beamfirings. This is followed by the Area/Volume Formation block, which canform an output point s_(out)(r _(ijk)) at location r _(ijk) using time,channel, and spatially dependent delay, apodization, phase, and summingacross N_(ch) channels, or some other suitable algorithm. In FIG. 8, theArea/Volume Formation block operates across channel data acquired fromN_(syn)(r _(ijk)) transmit beam firings, either applying a time,channel, and spatially dependent delay, apodization, phase, and summingacross N_(ch) channels, or other suitable algorithm, producing N_(syn)(r_(ijk)) outputs. Each of these outputs in turn are applied to theTransmit Synthesis block, consisting of N_(syn)(r _(ijk)) spatialcoordinate r _(ijk) dependent filters h^(syn) whose outputs are thensummed together across the N_(syn)(r _(ijk)) transmit beam firings toform an output point s_(out)(r _(ijk)) at location r _(ijk). Now sincethe received signals on each of the transducer elements provide arepresentation of the transmit beam pattern over the evolution oftransmit beam spatial firings, a spatial filter h^(syn), whose region ofsupport operates on either the channel signals, or area/volume formedsignals, across a finite number of the sequence of transmit beam spatialfirings which cover the region of interest, provides the desired spatialcompression through a properly delayed, phased, weighted, and filteredsuperposition. Therefore, the Transmit Synthesis block which performsthis filtering process can be viewed as a spatial diffraction transformsince it transforms the actual transmit diffraction response into adesired transmit diffraction response, whose characteristics, bothtemporal and spatial, can be substantially different.

While the transmit synthesis technique will be covered in more detail,at a high level, the basic idea is the following. Consider an expressionfor the round-trip beampattern formed from a sequence of transmit beamswhose apertures are centered at nΔxS _(n)(ω,x)=S _(xmt)(ω,x−nΔx)S _(rcv)(ω,x)  (11)where S_(xmt) is the transmit beampattern, S_(rcv) is the receivebeampattern, ω is the carrier frequency, x is the target's lateralposition, and

$\begin{matrix}{{s_{n}\left( {t,x} \right)} = {\frac{1}{2\pi}{\int{{S_{n}\left( {\omega,x} \right)}{P(\omega)}{\mathbb{e}}^{j\;\omega\; t}{\mathbb{d}\omega}}}}} & (12)\end{matrix}$where s_(n)(t,x) is the round-trip point spread function response forpulse spectrum P(ω). Forming a linear combination of the sequence ofroundtrip beampatterns, yields

$\begin{matrix}{{S^{syn}\left( {\omega,x} \right)} = {{\sum\limits_{n = 1}^{N_{syn}}\;{{H_{n}(\omega)}{S_{n}\left( {\omega,x} \right)}}} = {\sum\limits_{n = 1}^{N_{syn}}\;{{H_{n}(\omega)}{S_{xmt}\left( {\omega,{x - {n\;\Delta\; x}}} \right)}{S_{rcv}\left( {\omega,x} \right)}}}}} & (13)\end{matrix}$

This can be rewritten as

$\begin{matrix}{{S^{syn}\left( {\omega,x} \right)} = {{{S_{xmt}^{syn}\left( {\omega,x} \right)}{S_{rcv}\left( {\omega,x} \right)}\mspace{14mu}{S_{xmt}^{syn}\left( {\omega,x} \right)}} = {{\sum\limits_{n = 1}^{N_{syn}}\;{{H_{n}(\omega)}{S_{xmt}\left( {\omega,{x - {n\;\Delta\; x}}} \right)}}} = {\sum\limits_{n = 1}^{N_{syn}}\;{\underset{\underset{H_{n}{(\omega)}}{︸}}{{{H_{n}(\omega)}}{\mathbb{e}}^{j\;{\psi_{n}{(\omega)}}}}{S_{xmt}\left( {\omega,{x - {n\;\Delta\; x}}} \right)}}}}}} & (14)\end{matrix}$where H_(n)(ω) can be thought of as a diffraction correction filter(complex in general, i.e., magnitude/phase, real/imaginary, etc.), thefiltering operation termed diffraction transform. Thus, the operationattempts to transform, or correct, a sequence of actual transmitdiffraction patterns S_(xmt)(ω,x−nΔx), into the desired transmitdiffraction pattern S_(xmt) ^(syn) using N_(syn) zones.

The symbols in the above equations are defined as follows:

-   r_(xmt)=transmit focus range along the transmit beam direction-   r=point scatterer location in 2-D/3-D space-   r′=transducer array receive element location in 2-D/3-D space,    either electronically swept, mechanically swept, or a combination    thereof-   r _(ijk)=image/volume point location in 2-D/3-D space respectively-   φ(t)=demodulation phase-   s_(xmt)(t)=transmitted RF pulse waveform from transducer element-   {tilde over (s)}_(xmt)(t,r, . . . )=transmit beam RF pulse wavefront    at point scatterer location r due to transmit excitation s_(xmt)(t),    where ˜ indicates that the pulses transmitted by each element on the    array will interfere at the location r and therefore, the temporal    response of the pulse may be altered due to transmit diffraction-   {tilde over (s)}_(ch)(t, . . . )=receive channel signal from    transducer element following TGC, A/D conversion, demodulation,    baseband filtering, etc. ˜ indicates that the pulses received by    each element on the array may have been altered by transmit    diffraction-   T_(xmt)( . . . )=effective time delay of the transmit beam RF pulse    wavefront {tilde over (s)}_(xmt)(t,r, . . . ) to a point scatterer    located at r, i.e., the propagation delay plus the differential    delay arc of the transmit pulse wavefront, ΔT_(xmt), due to    diffraction-   T_(rcv)( . . . )=time delay of the backscattered pulse from a point    scatterer located at r to a transducer receive element-   T(ch,r _(ijk)), φ(ch,r _(ijk)), a(ch,r _(ijk))=receive channel delay    and phase correction, and apodization, used in area/volume formation    in 2-D/3-D space-   h_(rcv)(t)=receive filter which includes the effects of frequency    dependent attenuation, 1/r falloff, element response, the    transducer's one-way receive response, etc.-   h^(syn)( . . . )=transmit synthesis filter-   N_(syn)( . . . ), N _(syn)( . . . )=number/list of transmit beams in    2-D/3-D space used in transmit synthesis    Example: Planar 1-D Transducer, Linear Scan Format, 2-D Image    Formation

Consider the demodulated receive signal from a given transducer elementas shown in FIG. 6. The signal {tilde over (s)}_(ch) received by atransducer element located at r′=x′_(p), from a point scatterer atr=(r,x), insonified by the n-th transmit beam wavefront {tilde over(s)}_(xmt)(t−T_(xmt),x−x_(n),r,r_(xmt)), due to a transmit excitation ofs_(xmt)(t), where x_(n) is the transmit beam origin location on thearray, can be mathematically expressed as given by (15). The receivesignal s_(ch) is simply the backscattered signal from the pointscatterer at (r,x), attenuated and delayed by the propagation timeT_(rcv) from (r,x) back to the transducer receive element at x′_(p).

$\begin{matrix}{T_{xmt} = {{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)} = {\frac{r}{c} + {\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}}}}} & (15) \\{T_{rcv} = {{T_{rcv}\left( {{x_{p}^{\prime} - x},r} \right)} =}} & \; \\{\mspace{85mu}{{\frac{\sqrt{r^{2} + \left( {x_{p}^{\prime} - x} \right)^{2}}}{c}{{\overset{\sim}{s}}_{ch}\left( {{t - T_{xmt} - \mspace{11mu} T_{rcv}},x_{p}^{\prime},{x - x_{n}},r,r_{xmt}} \right)}} =}} & \; \\{\mspace{191mu}{{\mathbb{e}}^{{- j}\;{\varphi_{d}{(t)}}}{\int{{h_{rcv}\left( {{t - t^{\prime}},x_{p}^{\prime}} \right)}{{\overset{\sim}{s}}_{xmt}\left( {{t^{\prime} - T_{xmt} - T_{rcv}},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{d}t^{\prime}}}}}} & \;\end{matrix}$

If the transmit excitation by the transducer is represented bys _(xmt)(t)=p(t)e ^(jω) ^(x) ^(t)  (16)where p(t) is the transmit pulse envelope, potentially complex, andω_(x) is the transmit carrier frequency, then without loss of generalitythe transmit pulse wavefront can be expressed as{tilde over (s)} _(xmt)(t−T _(xmt) ,x−x _(n) ,r,r _(xmt))={tilde over(p)}(t−T _(xmt) ,x−x _(n) ,r,r _(xmt))e ^(jω) ^(x) ^((t−T) ^(xmt) ⁾ S_(xmt)(x−x _(n) ,r,r _(xmt),ω_(x))  (17)where ˜ indicates that the pulse envelope has been modified due to rangeand position dependent transmit diffraction effects, ω_(x) is theeffective transmit carrier frequency which may be depth dependent due tofrequency dependent attenuation effects, and S_(xmt) represents thetransmit beampattern. Upon substitution into equation (15), the receivechannel signal becomes{tilde over (s)} _(ch)(t−T _(xmt) −T _(rcv) ,x′ _(p) ,x−x _(n) ,r,r_(xmt))={tilde over (p)}_(rcv)(t−T _(xmt) −T _(rcv) ,x−x _(n) ,r,r_(xmt))·e ^(−jφ) ^(d) ^((t)+jω) ^(e) ^((t−T) ^(xmt) ^(−T) ^(rcv) ⁾ S_(xmt)(x−x _(n) ,r,r _(xmt),ω_(c))  (18)where {tilde over (p)}_(rcv)(t−T_(xmt)−T_(rcv),x−x_(n),r,r_(xmt))includes the effects of transmit diffraction, receive propagation backto the transducer, attenuation, filtering, etc., and ω_(c) is the depthdependent receive carrier frequency. Matching the demodulation andreceive carrier frequencies results in the baseband channel signal{tilde over (s)} _(ch)(t−T _(xmt) −T _(rcv) ,x′ _(p) ,x−x _(n) ,r,r_(xmt))={tilde over (p)}_(rcv)(t−T _(xmt) −T _(rcv) ,x−x _(n) ,r,r_(xmt))·e ^(−jω) ^(c) ^(T) ^(rcv) e ^(−jω) ^(c) ^(T) ^(xmt) S _(xmt)(x−x_(n) ,r,r _(xmt),ω_(c))  (19)

Image/Volume Reconstruction I shown in FIG. 7, creates a single channeldomain dataset from which the Area Formation block operates on after theTransmit Synthesis block has effectively compressed the transmit beam.Let the transmit synthesis and area formation outputs be given by

$\begin{matrix}{{{T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)} = {{\frac{r_{ij} + \sqrt{r_{ij}^{2} + \left( {x_{p}^{\prime} - x_{ij}} \right)^{2}}}{c}{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},{\underset{\_}{r}}_{ij},r,x,r_{xmt}} \right)}} = \underset{\underset{{Spatial}\mspace{14mu}{Diffraction}\mspace{14mu}{Transform}}{︸}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;{\int{{h_{n}^{syn}\left( {{t - t^{\prime}},x_{p}^{\prime},{\underset{\_}{r}}_{ij}} \right)}{{\overset{\sim}{s}}_{ch}\left( {{t^{\prime} - T_{xmt} - T_{rcv}},x_{p}^{\prime},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{d}t^{\prime}}}}}}}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} = \underset{\underset{{Area}\mspace{14mu}{Formation}}{︸}}{\sum\limits_{p}\;{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\delta\left\lbrack {t - {T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}} \right\rbrack}{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},{\underset{\_}{r}}_{ij},r,x,r_{xmt}} \right)}}}}} & (20)\end{matrix}$

Inserting equation (19) into equation (20) yields an expression for thesynthesized channel signals, namely

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},{\underset{\_}{r}}_{ij},r,x,r_{xmt}} \right)} = {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;{\int{{h_{n}^{syn}\left( {{t - t^{\prime}},x_{p}^{\prime},{\underset{\_}{r}}_{ij}} \right)}{{\overset{\sim}{P}}_{rcv}\left( {{t^{\prime} - T_{xmt} - T_{rcv}},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}T_{rcv}{\mathbb{e}}^{{- {j\omega}_{c}}T_{xmt}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}{\mathbb{d}t^{\prime}}}}}} & (21)\end{matrix}$

Consider three different functional forms for the transmit synthesisfilters h_(n) ^(syn), namely

$\begin{matrix}{{h_{n}^{syn}\left( {t,x_{p}^{\prime},{\underset{\_}{r}}_{ij}} \right)} = \left\{ \begin{matrix}{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta(t)}} \\{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta\left\lbrack {t + {\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}} \right\rbrack}} \\{h^{syn}\left\lbrack {{t + {\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}},{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right\rbrack}\end{matrix} \right.} & (22)\end{matrix}$

These transmit synthesis filter forms: a) are lateral convolutionfilters, and b) depend upon image range coordinate r_(ij) relative totransmit focus range r_(xmt) so as to provide range dependent transmitbeam diffraction correction. The N_(syn) filters of h^(syn) areapproximately centered around the n^(th) transmit beam closest to thelateral image coordinate x_(ij), and are designed in such a manner as tocorrect for the amplitude variations and range r dependent approximatequadratic delay/phase error experienced by the received backscatteredsignals as the transmit beam is translated across the region ofinterest, i.e., a spatial chirp compression filter. The number offilters necessary to provide complete spatial compression varies withdepth r and hence, image coordinate r_(ij), being dependent upon howwide the defocused transmit beam is, which is based upon how far r isfrom the transmit focus range r_(xmt). The wider the defocused transmitbeam, the greater the number of filters required for full spatialcompression. To the extent that a limited number of filters are used dueto fixed computational resources or otherwise, the transmit beam only bepartially compressed, which in many cases can suppress receive sidelobeclutter dramatically. For image coordinate locations r_(ij) around thetransmit focus range r_(xmt), not many filters are required since thetransmit beam is nearly focused.

The lateral convolution form of these transmit synthesis filtersprovides an effective lateral spatial shifting of the transmit beam,from the combination of spatial transmit beam firings, to center thesynthesized transmit beam over the image coordinate location x_(ij).This property effectively decouples transmit and round-trip spatialsampling requirements, while creating a transmit beam centered over thedesired image location. This property is achievable provided that thetransmit beams satisfy spatial Nyquist sampling requirements. Optimaldesign of the transmit synthesis filters requires sufficientlypredictable temporal and spatial pre-detection, i.e., coherent,characteristics of both the actual transmit beams and the desiredsynthesized transmit beams as they are swept through the region ofinterest, whether electronically or mechanically scanned, or anycombination thereof, as well as taking into account the number ofsynthesis filters N_(syn) used at a given image coordinate location. Ingeneral, the filtering of signals can yield different results dependingupon whether the signal of interest is coherent, i.e., backscatteredsignals from the region of interest, or incoherent in origin, i.e.,random noise, and produce changes in the signal to noise ratio (SNR).Since these transmit synthesis filters will process channel domain datafrom multiple transmit beams, each of which will contain receivefront-end random noise, optimal design of the transmit synthesis filtersmay also include these effects.

The first form of h_(n) ^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ(t)represents a range dependent, complex lateral filter, i.e.,magnitude/phase, real/imaginary, etc., which is independent of time andsimply provides an amplitude weighted phase adjustment of the transmitbeam dependent channel domain data sets prior to transmit synthesissummation. Referring to FIG. 7 which shows a representation of thereceive channel signals across transmit firings for a point scattererwhich lies shallow to the transmit focus range, i.e., r=r_(a)<r_(xmt),the backscattered receive signals arrive earlier as the transmit beam islaterally translated away from the position of the point scatterer at(r,x) due to the converging transmit pulse wavefront. The weighted phaseadjustment is sufficient provided that the delay excursion of thetransmit pulse wavefront, for the spatial portion of the defocusedtransmit beam to be compressed, or refocused, is on the order of theround-trip pulse length or less.

The second form of h_(n)^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ[t+ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt))]represents a range dependent, complex lateral filter as well; however,in addition to an amplitude weighted phase adjustment of the transmitbeam dependent channel domain data sets prior to transmit synthesissummation, it provides a differential time delay correctionΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)) (refer to equation (15)) to thebackscattered receive channel signals as the transmit beam is laterallytranslated away from the position of the point scatterer at (r,x) due tothe converging/diverging transmit pulse wavefront. This additional timecorrection will be required when the delay excursion of the transmitpulse wavefront, for the spatial portion of the defocused transmit beamto be compressed, or refocused, begins to exceed the round-trip pulselength.

The third form of h_(n)^(syn)=h^(syn)[t+ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)),x_(n)−x_(ij),r_(ij),r_(xmt)]represents a range dependent, complex lateral filter as well; however,in addition to an amplitude weighted phase adjustment and differentialtime delay correction ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)) applied tothe receive channel signals, the temporal filter portion also provides arange and lateral position dependent pulse shape correction. The intentis to compensate the receive channel signals {tilde over(s)}_(ch)(t−T_(xmt)−T_(rcv),x′_(p),x−x_(n),r,r_(xmt)), and throughextension, the transmit signal {tilde over(s)}_(xmt)(t−T_(xmt),x−x_(n),r,r_(xmt)), for diffraction effects awayfrom the transmit focus, receive filtering, etc., in addition todifferential transmit delay correction and transmit refocusing.

Substitution of the first form of h_(n)^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ(t) into the synthesizedchannel data given by equation (21), along with equation (15) yields

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} = {{\mathbb{e}}^{{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{P}}_{rcv}\left( {{t - T_{xmt} - T_{rcv}},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}}} & (23)\end{matrix}$

In order to determine a solution for the transmit synthesis filtercoefficients, they may be derived based upon theoretical considerations,i.e., deriving a spatial compression filter analogous to linear FM chirppulse compression, or an error function of the following form can beminimized at each image location (r_(ij),x_(ij)), over point scattererlateral locations x, using the list of potential range/positiondependent N _(syn)(r_(ij),x_(ij)) transmit beams, generally centeredabout the transmit beam closest to x_(ij), i.e., x₀(x_(ij)):

$\begin{matrix}{ɛ^{2} = {\sum\limits_{x}\;{❘{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{{\overset{\sim}{P}}_{rcv}\left\lbrack {{{- \Delta}\;{T_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt}} \right)}},{x - x_{n}},r_{ij},r_{xmt}} \right\rbrack}.{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}}{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}} \right\}} - {{p^{desired}\left( {0,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},{r_{ij}r_{xmt}},\omega_{c}} \right)}}}❘^{2}}}}} & (24)\end{matrix}$where

$p^{desired}\left( {{t - \frac{2r_{ij}}{c}},{x - x_{ij}},r_{ij}} \right)$and S_(xmt) ^(desired)( . . . ) are defined as the desired transmitpulse response and desired beampattern respectively, or takentogether—the desired transmit point spread function, which is thetargeted response for the spatial diffraction transform. t is evaluatedat the nominal receive time t=2r/c (see FIG. 6), r is evaluated at theimage range coordinate r_(ij), receive channel location x′_(p) isevaluated at the point scatterer location x. If S_(xmt) ^(desired)( . .. ) is that of a continuous focused transmit beam, then the subsequenttransmit synthesis filter coefficients derived through errorminimization or other means, will provide spatial compression of thetransmit beam away from the focus, in addition to the spatial shiftingproperty. Multiple time locations t can also be used in evaluating{tilde over (p)}_(rcv)( . . . ), S_(xmt)( . . . ), and e^(−jω) ^(c)^(ΔT) ^(xmt) in equation (23) in forming the error function, as well asin p^(desired)( . . . ). Other error functions are possible. For oneskilled in the art, there are many ways in which to perform this errorminimization, for example a least-squares, weighted least-squares,constrained least-squares error minimization, etc., where a set ofsimultaneous linear equations are created from the error function, andthe unknown coefficients h_(n) ^(syn)=h^(syn)(n) are solved. For a fixednumber of transmit synthesis filters N_(syn)(r_(ij),x_(ij)), errorminimization provides for a flexible solution. Evaluation of the receivepulse waveform(s) {tilde over (p)}_(rcv)( . . . ), transmit beampatternS_(xmt)( . . . ), desired pulse response p^(desired)( . . . ) andbeampattern S_(xmt) ^(desired)( . . . ), and transmit differential timedelay ΔT_(xmt)( . . . ) in the error minimization can be performedeither through experimental measurement, simulation, or a combinationthereof, provided they are sufficiently predictable. Applying theoptimized transmit synthesis filter coefficients to the receive channelsignals produces synthesized channel data given by

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} \approx {{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}}{\overset{\sim}{P}}_{rcv}\left\{ {{t - {\Delta\;{T_{xmt}\left\lbrack {x_{ij} - {x_{0}\left( x_{ij} \right)}} \right\rbrack}} - \left( {\frac{r}{c} + T_{rcv}} \right)},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}} & (25)\end{matrix}$where the synthesized transmit beampattern is given by

$\begin{matrix}{{s_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)} = {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (26)\end{matrix}$Note that the synthesized transmit beampattern will be centered over theimage coordinate location x_(ij) as described previously.

Upon substituting equation (25) into the expression for the AreaFormation block output given by equation (20), an expression is obtainedfor the round-trip point spread function

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} \approx {\sum\limits_{p}\;{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{{j\phi}{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}{{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}} \cdot {\overset{\sim}{p}}_{rcv}}\left\{ {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left\lbrack {x_{ij} - {x_{0}\left( x_{ij} \right)}} \right\rbrack}}},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}}} & (27)\end{matrix}$where the area formation element dependent delay T(x′_(p)−x_(ij),r_(ij))has effectively cancelled the element dependent receive delayT_(rcv)(x′_(p)−x,r) given in equation (15), for r _(ij)˜r. Then,

$\begin{matrix}{{\overset{\sim}{s}\left( {{r_{ij} - r},{x_{ij} - x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- {j\omega}_{c}}\frac{2r}{c}}{\overset{\sim}{p}}_{rcv}{\left\{ {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left\lbrack {x_{ij} - {x_{0}\left( x_{ij} \right)}} \right\rbrack}}},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\} \cdot {S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (28)\end{matrix}$where the round-trip point spread function's parameters are meant toindicate that it is in the form of a convolution kernel which could varyslowly throughout the imaging region defined by (r_(ij),x_(ij)). Thereceive beampattern S_(rcv)( . . . ) is given by

$\begin{matrix}{{s_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} = {\sum\limits_{p}\;{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{{j\phi}{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}{\mathbb{e}}^{- {{j\omega}_{c}{({T_{rcv} - \frac{r}{c}})}}}}}} & (29)\end{matrix}$

In order to correct for receive element phase errors due to thediffering propagation paths from the point scatterer back to thetransducer array, the receive phase is given by

$\begin{matrix}{{\phi\left( {r_{ij},{x_{p}^{\prime} - x_{ij}}} \right)} = {{\omega_{c}\left\lbrack {{T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)} - \frac{2r_{ij}}{c}} \right\rbrack} = {\omega_{c}\left\lbrack \frac{\sqrt{r_{ij}^{2} + \left( {x_{p}^{\prime} - x_{ij}} \right)^{2}} - r_{ij}}{c} \right\rbrack}}} & (30)\end{matrix}$Substituting equation (30) into equation (29) yields an expression forthe receive beampattern

$\begin{matrix}{{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} = {\sum\limits_{p}\;{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{{j\omega}_{c}{\lbrack{\frac{\sqrt{r_{ij}^{2} + {({x_{p}^{\prime} - x_{ij}})}^{2}} - r_{ij}}{c} - \frac{\sqrt{r^{2} + {({x_{p}^{\prime} - x})}^{2}} - r}{c}}\rbrack}}}}}{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} \approx {{\mathbb{e}}^{{- {j\omega}_{c}}\frac{1}{2c}\frac{{({x - x_{ij}})}^{2}}{r}}{\sum\limits_{p}{a\;\left( {u_{p},r_{ij}} \right){\mathbb{e}}^{{- {j\omega}_{c}}\frac{u^{2}}{2c}{({\frac{1}{r} - \frac{1}{r_{ij}}})}}{\mathbb{e}}^{{j\omega}_{c}\frac{{({x - x_{ij}})}u}{r_{ij}c}}}}}}} & (31)\end{matrix}$where a Taylor series expansion of the phase was used in theapproximation and yields the well known Fourier transform of the receiveapodization for r_(ij)=r, i.e., in the focal plane. Note the expressionfor the round-trip point spread function given by equation (28) is inthe form of a linear convolution kernel in both image coordinates, witha slow variation in characteristics due to the remaining parameterdependencies. The parameter dependencies uncorrected by the first formof the transmit synthesis filters equation (22) are: a) the uncorrectedspatially dependent differential transmit delay, and b) the transmitdiffraction and receive filtering effects on the transmit pulse. Theseare addressed with the second and third forms of the transmit synthesisfilters given by equation (22) respectively.

Substituting the second form of h_(n)^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ[t+ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt))]into the synthesized channel data given by equation (21), along withequation (15) yields

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} \approx {{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left( {{t - T_{rcv}},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}}} & (32)\end{matrix}$where the differential transmit delay is effectively cancelled forx_(ij)˜x. Solving for the transmit synthesis filter coefficients, thefollowing error equation is minimized for each image location(r_(ij),x_(ij)), over point scatterer lateral locations x as before.

$\begin{matrix}{ɛ^{2} = {\sum\limits_{x}\;{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left( {0,{x - x_{n}},r_{ij},r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}} \right\}} - {{p^{desired}\left( {0,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},r_{ij},r_{xmt},\omega_{c}} \right)}^{2}}}}}} & (33)\end{matrix}$where time t is evaluated at the nominal receive time t=2r/c (see FIG.6). Note that since the differential transmit delay ΔT_(xmt)( . . . )has been corrected, the pulse signals on a given receive channel arealigned in time, i.e., sampling along the transmit delay arc.

As before, multiple time locations can be used in constructing the errorfunction. Applying the optimized transmit synthesis filter coefficientsto the receive channel signals produces synthesized channel data givenby

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} \approx {{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}}{{\overset{\sim}{p}}_{rcv}\left\lbrack {t,{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\rbrack}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}} & (34)\end{matrix}$where the synthesized transmit beampattern is given by

$\begin{matrix}{{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)} = {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (35)\end{matrix}$

Note that even though the differential transmit delay has beencorrected, a phase error due to the transmit delay still exists due todemodulation being performed prior to the correction. Note further thatthe synthesized transmit beampattern is now centered over the imagecoordinate location x_(ij) as before.

Upon substituting equation (34) into the expression for the AreaFormation block output given by equation (20), an expression is obtainedfor the round-trip point spread function

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} \approx {\sum\limits_{p}\;{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{{j\phi}{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}{{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c}T_{rcv}})}}} \cdot {{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\rbrack}}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}}} & (36)\end{matrix}$where the Area Formation element dependent delay T(x′_(p)−x_(ij),r_(ij))has effectively cancelled the element dependent receive delayT_(rcv)(x′_(p)−x,r) given in equation (15), for r _(ij)˜r. Then,

$\begin{matrix}{{\overset{\sim}{s}\left( {{r_{ij} - r},{x_{ij} - x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- {j\omega}_{c}}\frac{2r}{c}}{{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\rbrack} \cdot {S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (37)\end{matrix}$where the receive beampattern S_(rcv)( . . . ) is the same as before,given by equations (29)-(31).

Substituting the third form of h_(n)^(syn)=n^(syn)[t+ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)),x_(n)−x_(ij),r_(ij),r_(xmt)]into the synthesized channel data given by equation (21), using variablesubstitution along with equation (15) yields

$\begin{matrix}{{{\overset{\sim}{s}}_{ch}^{syn}\left( {{t - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} \approx {{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}} \cdot {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{\int{{h^{syn}\left\lbrack {{t - t^{\prime}},{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right\rbrack}{{\overset{\sim}{p}}_{rcv}\left( {{\overset{\sim}{t} - T_{rcv}},{x - x_{n}}, r, r_{xmt}} \right)}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}{\mathbb{d}t^{\prime}}}}}}} & (38)\end{matrix}$

Solving for the transmit synthesis filter coefficients assuming a finitenumber of discrete time samples t_(m), m=[0,1, . . . M−1], the followingerror equation needs to be minimized, for each image location(r_(ij),x_(ij)), over point scatterer lateral locations x and time t

$\begin{matrix}{ɛ^{2} = {\sum\limits_{t}{\sum\limits_{x}\;{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\;\left\{ {\sum\limits_{m^{\prime} = 0}^{M - 1}{\left\{ {{h^{syn}\left( {{t - t_{m^{\prime}}},{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left( {t_{m^{\prime}},{x - x_{n}},r_{ij},r_{xmt}} \right)}} \right\}{\mathbb{e}}^{{- {j\omega}_{c}}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}}} \right\}} - {{p^{desired}\left( {t,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},r_{ij},r_{xmt},\omega_{c}} \right)}^{2}}}}}}} & (39)\end{matrix}$where time t′ is evaluated around the nominal receive time t′=2r/c (seeFIG. 6). Note that since the differential transmit delay ΔT_(xmt)( . . .) has been corrected, the pulse signals on a given receive channel arealigned in time, i.e., sampling along the transmit delay arc. Inaddition, the time dependence of the desired transmit pulse responsep^(desired)(t,x−x_(ij),r_(ij)) appears in the error minimizationfunction. A set of simultaneous linear equations are created from theerror function, and the unknown coefficients h_(mn) ^(syn)=h^(syn)(m,n)are solved. Thus, the transmit synthesis coefficients will attempt tocorrect the round-trip temporal response for transmit diffractioneffects as well as receive filtering. Applying the optimized transmitsynthesis filter coefficients to the receive channel signals producessynthesized channel data given by

$\begin{matrix}{{s_{ch}^{syn}\left( {{t - T_{rcv}},x_{p}^{\prime},x_{ij},r_{ij},x,r,r_{xmt}} \right)} \approx {{\mathbb{e}}^{- {{j\omega}_{c}{({\frac{r}{c} + T_{rcv}})}}}{p^{syn}\left( {t,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}} & (40)\end{matrix}$where the synthesized transmit point spread function is given by

$\begin{matrix}{{{p^{syn}\left( {t,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}} \approx {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{\sum\limits_{m^{\prime} = 0}^{M - 1}\;{\left\{ {{h^{syn}\left( {{t - t_{m^{\prime}}},{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left( {t_{m^{\prime}},{x - x_{n}},r,r_{xmt}} \right)}} \right\}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}}} & (41)\end{matrix}$

Upon substituting equation (40) into the expression for the AreaFormation block output given by equation (20), an expression is obtainedfor the round-trip point spread function

$\begin{matrix}{{s\left( {r_{ij},r,x_{ij},x} \right)} \approx {\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{\mathbb{e}}^{{- j}\;{\omega_{c}{({\frac{r}{c} + T_{rcv}})}}}{p^{syn}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{ij}},r_{ij}} \right\rbrack}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}}} & (42)\end{matrix}$where the Area Formation element dependent delay T(x′_(p)−x_(ij),r_(ij))has effectively cancelled the element dependent receive delayT_(rcv)(x′_(p)−x,r) given in equation (15), for r _(ij)˜r. Then,

$\begin{matrix}{{s\left( {{r_{ij} - r},{x_{ij} - x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2r}{c}}{p^{syn}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{ij}},r_{ij}} \right\rbrack}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (43)\end{matrix}$where the receive beampattern S_(rcv)( . . . ) is the same as before,given by equations (29)-(31). Thus, using the more general transmitsynthesis filter form, both differential time delay correction and pulseshape correction can be obtained simultaneously.

Image/Volume Reconstruction II shown in FIG. 8, which is a rearrangementof the Area Formation and Transmit Synthesis blocks, the Area Formationblock operates separately on channel domain datasets, from which theTransmit Synthesis block effectively compresses the transmit beam acrossmultiple area formed outputs. Let the area formation and transmitsynthesis outputs be given by

$\begin{matrix}\begin{matrix}{{T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)} = \frac{r_{ij} + \sqrt{r_{ij}^{2} + \left( {x_{p}^{\prime} - x_{ij}} \right)^{2}}}{c}} \\{{{\overset{\sim}{s}}_{n}\left( {r_{ij},r,x_{ij},x} \right)} = \underset{\underset{{Area}\mspace{14mu}{Formation}}{︸}}{\begin{matrix}{\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\delta\left\lbrack {t - {T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}} \right\rbrack}}} \\{{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{{\overset{\sim}{s}}_{ch}\left( {{t - T_{xmt} - T_{rcv}},x_{p}^{\prime},{x - x_{n}},r,r_{xmt}} \right)}}\end{matrix}}} \\{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} = \underset{\underset{{Spatial}\mspace{14mu}{Diffraction}\mspace{14mu}{Transform}}{︸}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{\sum\limits_{r_{ij}^{\prime}}{{h_{n}^{syn}\left( {{x_{n} - x_{ij}},{r_{ij} - r_{ij}^{\prime}},r_{ij}} \right)}{{\overset{\sim}{s}}_{n}\left( {r_{ij}^{\prime},r,x_{ij},x} \right)}}}}}\end{matrix} & (44)\end{matrix}$

Inserting equation (19) into equation (44) yields an expression for thearea formation outputs, namely

$\begin{matrix}{{{\overset{\sim}{s}}_{n}\left( {r_{ij},r,x_{ij},x} \right)} = {\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\delta\left\lbrack {t - {T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}} \right\rbrack}{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{{\overset{\sim}{p}}_{rcv}\left( {{t - T_{xmt} - T_{rcv}},{x - x_{n}},r,r_{xmt}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}T_{rcv}}{\mathbb{e}}^{{- j}\;\omega_{c}T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (45)\end{matrix}$Simplifying equation (45), along with equation (15) yields

$\begin{matrix}{{{\overset{\sim}{s}}_{n}\left( {r_{ij},r,x_{ij},x} \right)} \approx {\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{\mathbb{e}}^{{- j}\;{\omega_{c}{({\frac{r}{c} + T_{rcv}})}}}{{\overset{\sim}{p}}_{rcv}\left\lbrack {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}}},{x - x_{n}},r,r_{xmt}} \right\rbrack}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (46)\end{matrix}$where the area formation element dependent receive delayT(x′_(p)−x_(ij),r_(ij)) has effectively cancelled the element dependentreceive delay T_(rcv)(x′_(p)−x,r) given in equation (15), for r _(ij)˜r.Then,

$\begin{matrix}{{{\overset{\sim}{s}}_{n}\left( {r_{ij},r,x_{ij},x} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2r}{c}}{{\overset{\sim}{p}}_{rcv}\left\lbrack {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}}},{x - x_{n}},r,r_{xmt}} \right\rbrack}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}} & (47)\end{matrix}$where the receive beampattem S_(rcv)( . . . ) is given by

$\begin{matrix}{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} = {\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{j\;{\phi{({r_{ij},{x_{p}^{\prime} - x_{ij}}})}}}{\mathbb{e}}^{{- j}\;{\omega_{c}{({T_{rcv} - \frac{r}{c}})}}}}}} & (48)\end{matrix}$which is identical to equation (29). In order to correct for receiveelement phase errors due to the differing propagation paths from thepoint scatterer back to the transducer array, the receive phase is givenby equation (30) (repeated here for convenience)

$\begin{matrix}{{\phi\left( {r_{ij},{x_{p}^{\prime} - x_{ij}}} \right)} = {{\omega_{c}\left\lbrack {{T\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)} - \frac{2\; r_{ij}}{c}} \right\rbrack} = {\omega_{c}\left\lbrack \frac{\sqrt{r_{ij}^{2} + \left( {x_{p}^{\prime} - x_{ij}} \right)^{2}} - r_{ij}}{c} \right\rbrack}}} & (49)\end{matrix}$

Substituting equation (49) into equation (48) yields an expression forthe receive beampattern

$\begin{matrix}{{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} = {\sum\limits_{p}{{a\left( {{x_{p}^{\prime} - x_{ij}},r_{ij}} \right)}{\mathbb{e}}^{j\;{\omega_{c}\lbrack{\frac{\sqrt{r_{ij}^{2} + {({x_{p}^{\prime} - x_{ij}})}^{2}} - r_{ij}}{c} - \frac{\sqrt{r^{2} + {({x_{p}^{\prime} - x})}^{2}} - r}{c}}\rbrack}}}}}{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{1}{2c}\frac{{({x - x_{ij}})}^{2}}{r}}{\sum\limits_{p}{{a\left( {u_{p},r_{ij}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}\frac{u^{2}}{2c}{({\frac{1}{r} - \frac{1}{r_{ij}}})}}{\mathbb{e}}^{j\;\omega_{c}\frac{{({x - x_{ij}})}u}{r_{ij}c}}}}}}} & (50)\end{matrix}$where a Taylor series expansion of the phase was used in theapproximation and yields the well known Fourier transform of the receiveapodization for r_(ij)=r, i.e., in the focal plane.

Proceeding in an analogous manner as for Image/Volume Reconstruction I,consider three different functional forms for the transmit synthesisfilters namely

$\begin{matrix}{{h_{n}^{syn}\left( {\underset{\_}{r}}_{ij} \right)} = \left\{ \begin{matrix}{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta\left( r_{ij} \right)}} \\{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta\left\lbrack {r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}} \right\rbrack}} \\{h^{syn}\left\lbrack {{x_{n} - x_{ij}},{r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}},r_{xmt}} \right\rbrack}\end{matrix} \right.} & (51)\end{matrix}$

These transmit synthesis filter forms: a) are lateral convolutionfilters, and b) depend upon image range coordinate r_(ij) relative totransmit focus range r_(xmt) so as to provide range dependent transmitbeam diffraction correction. However, these filters, in contrast tothose given by equation (22), operate on area/volume formed data asopposed to channel data. As before, the N_(syn) filters of h^(syn) areapproximately centered around the n^(th) transmit beam closest to thelateral image coordinate x_(ij), and are designed in such a manner as tocorrect for the amplitude variations and range r dependent approximatequadratic delay/phase error experienced by the received backscatteredsignals as the transmit beam is translated across the region ofinterest, i.e., a spatial chirp compression filter. The number offilters necessary to provide complete spatial compression varies withdepth r and hence, image coordinate r_(ij), being dependent upon howwide the defocused transmit beam is, which is based upon how far r isfrom the transmit focus range r_(xmt). However, note that with theTransmit Synthesis block following the Area Formation block, thetransmit synthesis filters do not have the opportunity to correct forvariations on a receive channel basis since the filters operate onchannel summed outputs.

The lateral convolution form of these transmit synthesis filtersprovides an effective lateral spatial shifting of the transmit beam,from the combination of spatial transmit beam firings, to center thesynthesized transmit beam over the image coordinate location x_(ij).This property effectively decouples transmit and round-trip spatialsampling requirements, while creating a transmit beam centered over thedesired image location. This property is achievable provided that thetransmit beams satisfy spatial Nyquist sampling requirements. Optimaldesign of the transmit synthesis filters requires sufficientlypredictable temporal and spatial pre-detection, i.e., coherent,characteristics of both the actual transmit beams and the desiredsynthesized transmit beams as they are swept through the region ofinterest, whether electronically or mechanically scanned, or anycombination thereof, as well as taking into account the number ofsynthesis filters N_(syn) used at a given image coordinate location. Ingeneral, the filtering of signals can yield different results dependingupon whether the signal of interest is coherent, i.e., backscatteredsignals from the region of interest, or incoherent in origin, i.e.,random noise, and produce changes in the signal to noise ratio (SNR).Since these transmit synthesis filters will process area formed datafrom multiple transmit beams, each of which will contain receivefront-end random noise, optimal design of the transmit synthesis filtersmay also include these effects.

The first form of h_(n)^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ(r_(ij)), represents a rangedependent, complex lateral filter, i.e., magnitude/phase,real/imaginary, etc., simply provides an amplitude weighted phaseadjustment of the area formed data prior to transmit synthesissummation. Referring to FIG. 7 which shows a representation of thereceive channel signals across transmit firings for a point scattererwhich lies shallow to the transmit focus range, i.e., r=r_(a)<r_(xmt),the backscattered receive signals arrive earlier as the transmit beam islaterally translated away from the position of the point scatterer at(r,x) due to the converging transmit pulse wavefront. The weighted phaseadjustment is sufficient provided that the delay excursion of thetransmit pulse wavefront, for the spatial portion of the defocusedtransmit beam to be compressed, or refocused, is on the order of theround-trip pulse length or less. Due to the differential transmit delayexcursion, receive focusing errors may result due to the time dependentnature of the dynamic receive area formation or beamformation process.However, receive focusing errors will not be large if

$\frac{c}{2}\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}$is within the receive focusing depth of field limits. Since thedifferential transmit delay will not be corrected with this filter form,the area formed outputs will be delayed by the differential transmitdelay—a range displacement error of

$\frac{c}{2}\Delta\;{{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}.}$

The second form of

${h_{n}^{syn} = {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta\left\lbrack {r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}} \right\rbrack}}},$represents a range dependent, complex lateral filter as well; however,in addition to an amplitude weighted phase adjustment of the transmitbeam dependent area formed data prior to transmit synthesis summation,it includes a range displacement correction

$\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}$(refer to equation (15)) to the backscattered receive channel signals asthe transmit beam is laterally translated away from the position of thepoint scatterer at (r,x) due to the converging/diverging transmit pulsewavefront. This additional range displacement correction will berequired when the delay excursion of the transmit pulse wavefront, forthe spatial portion of the defocused transmit beam to be compressed, orrefocused, begins to exceed the round-trip pulse length. Note that whilethe range displacement errors are corrected, any receive focusing errorsare not.

The third form of

${h_{n}^{syn} = {h^{syn}\left\lbrack {{x_{n} - x_{ij}},{r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}},r_{xmt}} \right\rbrack}},$represents a range dependent, complex lateral filter as well; however,in addition to an amplitude weighted phase adjustment, it includes arange displacement correction of

$\frac{c}{2}\Delta\;{{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}.}$The transmit synthesis filter also provides a range and lateral positiondependent pulse shape correction, however, operating on the area formedoutput range samples. The intent is to compensate the area formedoutputs and through extension, the receive channel signals {tilde over(s)}_(ch)(t−T_(xmt)−T_(rcv),x′_(p),x−x_(n),r,r_(xmt)), as well as thetransmit signal {tilde over (s)}_(xmt)(t−T_(xmt),x−x_(n),r,r_(xmt)), fordiffraction effects away from the transmit focus, receive filtering,etc., in addition to range displacement correction and transmitrefocusing. Note that while the range displacement errors are corrected,any receive focusing errors are not.

Substitution of the first form of h_(n)^(syn)=h^(syn)(x_(n)−x_(ij),r_(ij),r_(xmt))δ(r_(ij)) into the transmitsynthesis output given by equation (44), using the area formed outputsgiven by equation (47), and along with equation (15) yields

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} = {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)} \cdot {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left\lbrack {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}}},{x - x_{n}},r,r_{xmt}} \right\rbrack}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}} \right\}}}}} & (52)\end{matrix}$

Note that for the image range location at which the target will appear,i.e.,

${r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x - x_{n}},r,r_{xmt}} \right)}}},$the receive beampattern may contain a focusing error under theconditions described previously, and will be discussed in regard to thesecond transmit synthesis filter form. In order to determine a solutionfor the transmit synthesis filter coefficients, they may be derivedbased upon theoretical considerations, or an error function of thefollowing form can be minimized, for each image location(r_(ij),x_(ij)), over point scatterer lateral locations x, using thelist of potential range/position dependent N _(syn)(r_(ij),x_(ij))transmit beams, generally centered about the transmit beam closest tox_(ij), i.e., x₀(x_(ij)):

$\begin{matrix}{ɛ^{2} = {\sum\limits_{x}{{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{{\overset{\sim}{p}}_{rcv}\left\lbrack {{{- \Delta}\;{T_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt}} \right)}},{x - x_{n}},r_{ij},r_{xmt}} \right\rbrack} \cdot {\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}}{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}} \right\}} - {{p^{desired}\left( {0,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},r_{ij},r_{xmt},\omega_{c}} \right)}}}}2}}} & (53)\end{matrix}$where

$p^{desired}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{ij}},r_{ij}} \right\rbrack$and S_(xmt) ^(desired)( . . . ) are defined as the desired transmitpulse response and desired beampattern respectively, or takentogether—the desired transmit point spread function, which is thetargeted response for the spatial diffraction transform. r is evaluatedat the image range coordinate r_(ij). If S_(xmt) ^(desired)( . . . ) isthat of a continuous focused transmit beam, then the subsequent transmitsynthesis filter coefficients derived through error minimization orother means, will provide spatial compression of the transmit beam awayfrom the focus, in addition to the spatial shifting property. Multipleimage range locations r_(ij) can also be used in evaluating {tilde over(p)}_(rcv)( . . . ), S_(xmt)( . . . ), and e^(−jω) ^(c) ^(ΔT) ^(xmt) inequation (52), as well as p^(desired)( . . . ) in forming the errorfunction. Other error functions are possible. Evaluation of the receivepulse waveform(s) {tilde over (p)}_(rcv)( . . . ), transmit beampatternS_(xmt)( . . . ), desired pulse response p^(desired)( . . . ) andbeampattern S_(xmt) ^(desired)( . . . ), and transmit differential timedelay ΔT_(xmt)( . . . ) in the error minimization can be performedeither through experimental measurement, simulation, or a combinationthereof, provided they are sufficiently predictable. Applying theoptimized transmit synthesis filter coefficients to the receive areaformed output produces the round-trip point spread function

$\begin{matrix}{{\overset{\sim}{s}\left( {{r_{ij} - r},{x_{ij} - x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{\overset{\sim}{p}}_{rcv}{\left\{ {{\frac{2\left( {r_{ij} - r} \right)}{c} - {\Delta\;{T_{xmt}\left\lbrack {x_{ij} - {x_{0}\left( x_{ij} \right)}} \right\rbrack}}},{x - {x_{0}\left( x_{ij} \right)}},r,r_{xmt}} \right\} \cdot {S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (54)\end{matrix}$which contains an approximate range displacement error of

${\frac{c}{2}\Delta\;{T_{xmt}\left\lbrack {x_{ij} - {x_{0}\left( x_{ij} \right)}} \right\rbrack}},$the round-trip point spread function's parameters are meant to indicatethat it is in the form of a convolution kernel which could vary slowlythroughout the imaging region, and where the synthesized transmitbeampattern is given by

$\begin{matrix}{{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)} = {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (55)\end{matrix}$Note that the synthesized transmit beampattern is now centered over theimage coordinate location x_(ij) as described previously.

Substitution of the second form of

$h_{n}^{syn} = {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{\delta\left\lbrack {r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}} \right\rbrack}}$into the transmit synthesis output given by equation (40), using thearea formed outputs given by equation (47), and along with equation (15)yields

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} = {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2r}{c}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}^{\;}\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack} \cdot {\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}{S_{rcv}\left\lbrack {{x - x_{ij}},{r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}},r,\omega_{c}} \right\rbrack}} \right\}}}} & (56)\end{matrix}$

Note that while the range displacement error has been corrected—thetransmit differential delay term no longer appears in the receive pulseresponse, the dependence of the synthesized round-trip point spreadfunction {tilde over (s)}( . . . ) on the receive beampattern is nowtransmit beam index n dependent due to the range displacement term

$\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}$and introduces a receive focusing error given a sufficiently largedifferential transmit delay ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)). Thiscan seen by evaluating the receive beampattern S_(rcv)( . . . ) given byequation (50) at

$r_{ij} + {\frac{c}{2}\Delta\;{{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}.}}$Evaluating r=r_(ij), it produces the phase error term

$\begin{matrix}{\mathbb{e}}^{{- j}\;\omega_{c}{\frac{u^{2}}{2\; c}{\lbrack{\frac{1}{r_{ij}} - \frac{1}{r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}{({{x_{n} - x_{ij}},r_{ij},r_{xmt}})}}}}}\rbrack}}} & (57)\end{matrix}$

If sufficiently large, this quadratic phase error term will lead toreceive defocusing, which will also introduce a gain and phasemodification of the receive beampattern contribution to equation (56).Whether or not the receive beampattern dependence needs to be includedwithin the transmit synthesis coefficient solution depends upon whetherthe time shifts introduced by the differential transmit delay satisfythe following relationship.

$\begin{matrix}{\begin{matrix}{{\left. R_{DOF}^{rcv} \right.\sim\beta}\;\lambda\; F_{{rcv}^{\prime}}^{2}} & {4 \leq \beta \leq 8}\end{matrix}{{{\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}} \leq \frac{R_{DOF}^{rcv}}{2}}} & (58)\end{matrix}$where the choice of β depends upon what phase error is assumed at theend elements of the receive aperture in the depth of field (DOF)derivation. If the inequality in equation (58) is satisfied for imagelocations (r_(ij),x_(ij)) and transmit beam location x_(n) over transmitsynthesis filters N _(syn)(r_(ij),x_(ij)), then the receive beampatterncan be rearranged within the round-trip point spread function and isgiven by

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{{S_{rcv}\left\lbrack {{x - x_{ij}},r_{ij},r,\omega_{c}} \right\rbrack} \cdot {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - {x_{n}r}},r_{xmt},\omega_{c}} \right)}}}}}} & (59)\end{matrix}$from which the error function can be written as

$\begin{matrix}{ɛ^{2} \approx {\sum\limits_{x}{{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left( {0,{x - x_{n}},r_{ij},r_{xmt}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}\;{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}} \right\}} - {{p^{desired}\left( {0,{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},r_{ij},r_{xmt},\omega_{c}} \right)}}}}2}}} & (60)\end{matrix}$where r is evaluated at the image range coordinate r_(ij). Note thatsince the differential transmit delay ΔT_(xmt)( . . . ) has beencorrected, there is no range displacement errors. As before, multipleimage range locations r_(ij) can be used in the error minimization. Ifthe inequality in equation (58) is not satisfied, then the receivebeampattern should be included in the transmit synthesis coefficientcalculation procedure and/or error minimization. Applying the optimizedtransmit synthesis filter coefficients to the receive area formed outputproduces a round-trip point spread function given by

$\begin{matrix}{{\overset{\sim}{s}\left( {{r_{ij} - r},{x_{ij} - x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack} \cdot {S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (61)\end{matrix}$where the synthesized transmit beampattern is given by

$\begin{matrix}{{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)} = {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{{h^{syn}\left( {{x_{n} - x_{ij}}, r_{ij}, r_{xmt}} \right)}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}} & (62)\end{matrix}$

Substitution of the third form of

$h_{n}^{syn} = {h^{syn}\left\lbrack {{x_{n} - x_{ij}},{r_{ij} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}}},r_{xmt}} \right\rbrack}$into the transmit synthesis output given by equation (44), using thearea formed outputs given by equation (47), and along with equation (15)yields

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} = {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {\sum\limits_{r_{ij}^{\prime}}{\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},{r_{ij} - r_{ij}^{\prime}},r_{ij},r_{xmt}} \right)}{{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij}^{\prime} - r} \right)}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack} \cdot {s_{rcv}\left\lbrack {{x - x_{ij}},{{r\;}_{ij}^{\prime} + {\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij}^{\prime},r_{xmt}} \right)}}},r,\omega_{c}} \right\rbrack}}} \right\}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}} \right\}}}} & (63)\end{matrix}$where the extra r_(ij) dependence in h^(syn)( . . . ) is meant toindicate that the filter's range convolution kernel may change withimage range r_(ij) to compensate for the slow range dependence inherentin {tilde over (p)}_(rcv)[ . . . , r,r_(xmt)]. Note that while the rangedisplacement error has been corrected—the transmit differential delayterm no longer appears in the receive pulse response, the dependence ofthe synthesized round-trip point spread function {tilde over (s)}( . . .) on the receive beampattern is now transmit beam index n dependent dueto the range displacement term

$\frac{c}{2}\Delta\;{T_{xmt}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}$and introduces a receive focusing error given a sufficiently largedifferential transmit delay ΔT_(xmt)(x_(n)−x_(ij),r_(ij),r_(xmt)) asdiscussed previously. If the inequality in equation (58) is satisfiedfor image locations (r_(ij),x_(ij)) and transmit beam location x_(n)over transmit synthesis filters N _(syn)(r_(ij),x_(ij)), and the rangevariation r_(ij) of the receive beam pattern S_(rcv)( . . . ) is slowcompared with the time/range variation of

${{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},\ldots} \right\rbrack},$which is generally the case unless long pulses combined with low receiveF-numbers are used, then the receive beampattern can be rearrangedwithin the synthesized round-trip point spread function and is given by

$\begin{matrix}{{\overset{\sim}{s}\left( {r_{ij},r,x_{ij},x} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{{S_{rcv}\left\lbrack {{x - x_{ij}},r_{ij},r,\omega_{c}} \right\rbrack} \cdot {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{\sum\limits_{r_{ij}^{\prime}}{\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},{r_{ij} - r_{ij}^{\prime}},r_{ij},r_{xmt}} \right)}{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\left( {r_{ij}^{\prime} - r} \right)}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack}} \right\}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\; T_{xmt}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}}}}} & (64)\end{matrix}$

Solving for the transmit synthesis filter coefficients assuming a finitenumber of range samples r_(m), m=[0,1, . . . M-1], evaluated around thenominal receive range r. The following error equation needs to beminimized, for each image location (r_(ij),x_(ij)), over point scattererlateral locations x and fast range/time {circumflex over (r)}_(ij).

$\begin{matrix}{ɛ^{2} = {\sum\limits_{{\hat{r}}_{ij}}{\sum\limits_{x}{{{{\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}\left\{ {\sum\limits_{m^{\prime} = 0}^{M - 1}{\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},{{\hat{r}}_{ij} - r_{m^{\prime}}},r_{xmt}} \right)}\mspace{31mu}{{\overset{\sim}{p}}_{rcv}\left\lbrack {\frac{2\; r_{m^{\prime}}}{c},{x - x_{n}},r_{ij},r_{xmt}} \right\rbrack}} \right\}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r_{ij},r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r_{ij},r_{xmt},\omega_{c}} \right)}}} \right\}} - {{p^{desired}\left( {\frac{2\;{\hat{r}}_{ij}}{c},{x - x_{ij}},r_{ij}} \right)}{S_{xmt}^{desired}\left( {{x - x_{ij}},r_{ij},\; r_{xmt},\omega_{c}} \right)}}}}2}}}} & (65)\end{matrix}$

Note that since the differential transmit delay ΔT_(xmt)( . . . ) hasbeen corrected, there is no range displacement errors. In addition, therange/time dependence of the desired transmit pulse response

$p^{desired}\left( {\frac{2{\hat{r}}_{ij}}{c},{x - x_{ij}},r_{ij}} \right)$appears in the error minimization function. A set of simultaneous linearequations are created from the error function, and the unknowncoefficients h_(mn) ^(syn)=h^(syn)(m,n) are solved. If the inequality inequation (58) is not satisfied, or if long pulses are used incombination with low receive F-numbers, then the receive beampatternshould be included in the transmit synthesis coefficient calculationprocedure and/or error minimization. Applying the optimized transmitsynthesis filter coefficients to the receive area formed output producesa synthesized round-trip point spread function given by

$\begin{matrix}{{s\left( {{r_{ij} - r},{x_{ij}x},r_{ij},x_{ij}} \right)} \approx {{\mathbb{e}}^{{- j}\;\omega_{c}\frac{2\; r}{c}}{p^{syn}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{ij}},r_{ij}} \right\rbrack}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{x\;{mt}},\omega_{c}} \right)}{S_{rcv}\left( {{x - x_{ij}},r_{ij},r,\omega_{c}} \right)}}} & (66)\end{matrix}$where the synthesized transmit point spread function is given by

$\begin{matrix}{{{p^{syn}\left\lbrack {\frac{2\left( {r_{ij} - r} \right)}{c},{x - x_{ij}},r_{ij}} \right\rbrack}{S_{xmt}^{syn}\left( {{x - x_{ij}},r_{ij},r,r_{xmt},\omega_{c}} \right)}} \approx {\sum\limits_{n = {{\underset{\_}{N}}_{syn}{({\underset{\_}{r}}_{ij})}}}{\sum\limits_{m^{\prime} = 0}^{M - 1}{\left\{ {{h^{syn}\left( {{x_{n} - x_{ij}},r_{ij},r_{xmt}} \right)}{{\hat{p}}_{rcv}\left\lbrack {\frac{2\; r_{m^{\prime}}^{\prime}}{c},{x - x_{n}},r,r_{xmt}} \right\rbrack}} \right\}{\mathbb{e}}^{{- j}\;\omega_{c}\Delta\;{T_{xmt}{({{x - x_{n}},r,r_{xmt}})}}}{S_{xmt}\left( {{x - x_{n}},r,r_{xmt},\omega_{c}} \right)}}}}} & (67)\end{matrix}$

In the preceding analysis for both Image/Volume Reconstruction I and II,after obtaining the round-trip point spread function s( . . . ),assuming that linearity holds throughout the transmission, propagation,reception process, the pre-detected output s_(out)( . . . ) can be foundthrough the convolution of the round-trip point spread function with theregion of interest scatterer density given bys _(out)( r _(ijk))=∫ρ( r′)s( r _(ijk) −r′,r _(ijk))dr′  (68)where ρ(r) is the scatterer density.Image/Volume Reconstruction I & II Hybrid

In both Image/Volume Reconstruction I & II architectures, additionalinformation is being extracted from multiple channel data sets about thetransmit beam's temporal and spatial properties, from the evolution ofthe transmit beam as it is swept through the region of interest, whichprovides the opportunity to refocus the transmit beam at all spatiallocations, compensating for diffraction effects. Depending upon theparticular implementation and desired requirements, either type I or II,or a hybrid of types I and II may be more suitable. For example, toavoid potential receive focusing errors as described in Image/VolumeReconstruction II, represented by FIG. 8, the receive dynamic delay termcan include a transmit beam n and image coordinate dependentdifferential transmit delay correction in the Area/Volume Formationblock directly. This creates a hybrid between the two architecturesdescribed, and may have other advantages in a particular implementation.Other rearrangements between the Transmit Synthesis and Area/VolumeFormation blocks are possible.

Non-linear Imaging Modes

Both Image/Volume Reconstruction architectures and techniques developedare able to synthesize a transmit beam of desired temporal and spatialcharacteristics, from a collection of actual transmit beams whichpossess potentially different temporal and spatial characteristics,thereby preserving the beneficial effects the actual transmit beams mayhave had in generating desirable imaging characteristics, while at thesame time correcting for the undesirable characteristics after the facton receive. This is especially important for non-linear imaging modessuch as 2^(nd) harmonic imaging where the beneficial effects of clutterreduction come about due to the effective “squaring” of the transmitbeam pattern which provides reduction of the transmit mainlobe width,lowering of the transmit beam clutter level, etc., subsequent to bodyaberrations. As stated previously, other techniques which seek to breakup the transmit beam/aperture into several, or many, sequentialcomponents degrade this ability. While the examples provided assumedlinearity throughout the analysis, non-linear imaging modes such as2^(nd) harmonic imaging, etc., can be handled in a similar mannerassuming that the scattering process is approximately linear. Since thetransmit synthesis process occurs on the receive side following anynon-linear transmit propagation, all that is required is an expressionappropriate for describing the transmit pulse wavefront as it impingesupon the scatterer for the non-linear imaging mode of interest andsimply inserting it into equation (17) instead of the one appropriatefor linear, fundamental imaging. Alternatively, the transmit pulsewavefront for the non-linear imaging mode of interest can be expressedin terms of the linear, fundamental transmit pulse wavefront, eitherapproximately or otherwise. For example, a reasonable model for 2^(nd)harmonic imaging might be{tilde over (s)} _(xmt)(t−T _(xmt) ,x−x _(n) ,r,r _(xmt))={tilde over(p)} ²(t−T _(xmt) ,x−x _(n) ,r,r _(xmt))e ^(j2ω) ^(x) ^((t−T) ^(xmt) ⁾ S_(xmt) ²(x−x _(n) ,r,r _(xmt),ω_(x))  (69)The evaluation of the non-linearly generated transmit beam pattern andpulse response in the subsequent transmit synthesis error minimizationcan be performed using measurements, simulations, theoreticalconsiderations, or any combination thereof.Computation Efficiency

The computation efficiency of the transmit synthesis technique describedhas advantages over other techniques which seek to provide a dynamictransmit focus. Consider for example, one of the techniques brieflydescribed, namely transmit sub-apertures which breaks up the transmitaperture into many sub-apertures, which in the limit becomes that of asingle transmit element. For each image point, the resulting receivechannel signals are processed using delay, phase, apodization, andsummation across transmit channels or firings, prior to delay, phase,apodization, and summation across receive channels. In general, tomaintain constant resolution either on transmit or receive, the transmitand receive apertures need to increase as the focusing range increases.This implies that the number of computations involved, using operationssuch as delay, phase, and apodization in forming an image point,increases with the aperture size. For the sub-aperture technique, asshown in the left hand column of plots in FIG. 9, the transmit andreceive aperture size increase linearly with depth r, with the productof the two being proportional to the number of operations needed tocompute an image point at a particular depth r, which is given by aquadratic function.

The transmit synthesis technique described by the present invention cansynthesize a dynamic transmit focus from a collection of conventionallyfocused transmit beams. The amount of spatial compression and hence,number of transmit synthesis filters N^(syn) required to completelyfocus the transmit beam, is proportional to the range dependent width ofthe actual transmit beam. If the actual focus range is located towardsthe lower portion of the region of interest, which is typically thecase, the number of transmit synthesis filters required is larger in thenear field due to the defocused transmit beam width, decreasing withdepth as the focus range r_(xmt) is approached. At the focus ranger_(xmt), very few transmit synthesis filters are required, 2 or 3suffice since the transmit beam is already focused, and the synthesisfilters essentially provide the required spatial shifting of thetransmit beam. Considering the spatial compression aspects only, theright hand column of plots in FIG. 9 show the number of computationsinvolved in transmit synthesis and receive processing, with the productof the two being proportional to the number of operations needed tosynthesize an image point at a particular depth r, which is also givenby a quadratic function of a different shape (in the planar arrayexample, the transmit synthesis filters used similar operations, namelydelay, phase, and amplitude).

For the transmit sub-aperture technique, the number of operationscontinues to increase quadratically since the number of operationsneeded on transmit and receive both increase linearly with depth. Incontrast, the transmit synthesis technique of the present invention,which uses more operations to synthesize a focused transmit beam in thenear field and less operations to focus receive channels, and uses lessoperations to synthesize a focused transmit beam in the far field andmore operations to focus receive channels, provides a better balancingof processing resources as shown in FIG. 9. If the total number ofoperations is computed by integrating the range dependent number ofoperations, shown by the patterned areas in the lower plots of FIG. 9,and if the range dependent transmit and receive apertures, along withthe range dependent number of transmit synthesis filters areapproximately given by

$\begin{matrix}{{{\left. {N_{ch}^{xmt}(r)} \right.\sim\frac{1}{F_{xmt}d}}r}{{\left. {N^{syn}(r)} \right.\sim\frac{1}{F_{xmt}d}}\left( {r_{xmt} - r} \right)}{{\left. {N_{ch}^{sub}(r)} \right.\sim\frac{1}{F_{rcv}d}}r}{{\left. {N_{ch}^{rcv}(r)} \right.\sim\frac{1}{F_{rcv}d}}r}{{\left. {N_{ops}^{sub}(r)} \right.\sim\frac{1}{F_{xmt}F_{rcv}d^{2}}}r^{2}}{{\left. {N_{ops}^{sub}(r)} \right.\sim\frac{1}{F_{xmt}F_{rcv}d^{2}}}{r\left( {r_{xmt} - r} \right)}}{{\left. N_{ops}^{sub} \right.\sim{\int_{0}^{r_{xmt}}{{N_{ops}^{sub}(r)}{\mathbb{d}r}}}} = {\frac{1}{3\; F_{xmt}F_{rcv}s^{2}}r_{xmt}^{2}}}{{\left. N_{ops}^{syn} \right.\sim{\int_{0}^{r_{xmt}}{{N_{ops}^{syn}(r)}{\mathbb{d}r}}}} = {\frac{1}{6\; F_{xmt}F_{rcv}d^{2}}r_{xmt}^{3}}}} & (70)\end{matrix}$where d represents the transducer element spacing, then the ratio ofcomputation effort for the present invention versus the sub-aperturetechnique is approximately given by

$\begin{matrix}{{\left. \frac{N_{ops}^{syn}}{N_{ops}^{sub}} \right.\sim\frac{\left( {\frac{1}{6\; F_{xmt}F_{rcv}d^{2}}r_{xmt}^{3}} \right)}{\left( {\frac{1}{2\; F_{xmt}F_{rcv}d^{2}}r_{xmt}^{3}} \right)}}\mspace{50mu}{\left. \frac{N_{ops}^{syn}}{N_{ops}^{sub}} \right.\sim\frac{1}{2}}} & (71)\end{matrix}$The transmit synthesis technique described by the present inventionneeds approximately 50% fewer operations relative to the number ofoperations for the transmit sub-aperture technique.Signal to Noise Ratio (SNR)

Since the transmit synthesis technique described by the presentinvention combines multiple sets of signals, i.e., channel data,area/volume formed data, beamformed data, etc., providing spatialcompression to the diffracted transmit beam signals forming a continuousfocused transmit beam, receive front-end random noise will be affectedin a similar manner as in pulse compression techniques. In pulsecompression techniques, the SNR is improved the most where the pulse hasbeen the most dispersed. This implies that where the transmit beam isthe most defocused, the greatest potential gain in SNR can be achieved,which will occur at depths shallow to and deeper than, the actualtransmit focus. Around the transmit focus, the transmit beam will bemodified to a much lesser degree, if at all, maintaining the SNR(dependent upon the particular design of the transmit synthesisfilters). Thus, the SNR after transmit synthesis, whether performed onchannel data or area formed data, can display an improvement as afunction of depth as shown in FIG. 10. The SNR improvement usingtransmit synthesis performed in three dimensions, i.e., volumeformation, should be even greater due to the added spatial compressiondimension, i.e., elevation.

Scan Formats, 3-D Imaging Modes

The Reconstruction topologies and techniques developed by the presentinvention are able to synthesize a transmit beam of desired temporal andspatial characteristics, from a collection of actual transmit beamswhich possess potentially different temporal and spatialcharacteristics, where the collection of actual transmit beams { . . .n−1 n n+1 . . . } are swept through the region of interest in anarbitrary manner in 3-D space, either mechanically, electronically, orany combination thereof, making it compatible with all forms of scanningformats, i.e., linear, sector, curved, etc., employing transducers ofarbitrary array geometries. The transducers employed can be 1-D arrays,1-1/2-D arrays, or 2-D arrays. The transmit synthesis compression canoccur in an arbitrary 2-D plane, or 3-D space. All the precedinganalysis is still valid—the image/volume coordinates used in theanalysis are simply changed to ones appropriate for the describingspatial locations in 3-D space. The transmit synthesis errorminimization may now contain additional spatial dimensions foroptimization. For example, in the case of a linear scan format in bothazimuth x and elevation y directions, the error minimization procedureto solve for the transmit synthesis filters now contains the addedevaluation of the elevation beampattern along the y direction, inaddition to the lateral x evaluation of the azimuth transmitbeampattern, etc.

Motion Effects

In general, whenever signals are combined from multiple transmitfirings, the potential exists for effects due to motion to alter theresulting signal combination, whether the signals are pre-detected orpost-detected in origin. However, typically it is the case thatpre-detected, or coherent, combination of signals will display greaterchanges due to the interference between the signals resulting frommotion induced phase shifts. Since the present invention using transmitsynthesis combines coherent signals from multiple transmit firings, theresulting spatial compression can be influenced by motion. Generally,the greater the number of signals combined from multiple transmitfirings, the greater the potential sensitivity to motion. To the extentthat motion induced effects are undesirable in the spatial compressionprocess, which may not always be the case, several approaches can beemployed to address motion effects. One approach is to change the amountof spatial compression employed, which is proportional to the number oftransmit synthesis filters h_(n) ^(syn)( . . . ), which can be reducedaccordingly, whether a priori in the design, or adaptive to the motionitself. A diagram of this process is shown in FIG. 11A where the MotionProcessor block detects the motion from firing to firing employingchannel data, and then computes appropriate parameters for the TransmitSynthesis block in order to reduce the sensitivity to motion such asthrough reduced number of transmit synthesis filters N^(syn), filterdefinition h_(n) ^(syn)( . . . ), etc. The distinct advantage thepresent invention has over some other techniques such as transmitsub-apertures, is that the actual transmit beam used in data acquisitionhas already been formed in space, unaltered due to motion. Many transmitfirings are not required to synthesize a transmit beam. The transmitsynthesis process of the present invention spatially compresses thetransmit beam away from the actual transmit focus, as well as aligningthe transmit beam over the imaging point, to the extent that a)computational resources, i.e., number of transmit synthesis filters, areused to combine multiple coherent datasets, and b) motion, permit.

A second approach to address motion effects is to simply correct thedata for motion induced spatial shifts in an adaptive manner. This canbe accomplished through a variety of techniques well known in the fieldsuch as spatial correlation, either in 2-D or 3-D space. The datasamples, either channel data or area/volume formed data, are spatiallyshifted and/or interpolated in an image, or volume position dependentmanner, including any phase corrections for motion induced phase shifts,etc., prior to transmit synthesis processing. Thus any motioncompensation processing would occur prior to the Transmit Synthesisblock, or prior to the Reconstruction block, etc. This can also berepresented by FIG. 11A where in this case, the Motion Processor blockpasses on motion corrected channel data to the Transmit Synthesis block.Note that the Transmit Synthesis block and Area/Volume Formation blockscan be interchanged, or the combination of the two modified as mentionedpreviously, in which case the Motion Processor block in FIG. 11A may beplaced following the Area/Volume Formation block and prior to theTransmit Synthesis block. Other rearrangements are possible.

A third approach to address motion effects, which is also an adaptivetechnique, is to perform multiple transmit synthesis processing inparallel, using different sets of transmit synthesis filter designsh_(n) ^(syn)( . . . ), and combining each of the transmit synthesisoutputs in a adaptive manner. For example, FIG. 11B and FIG. 11Crepresent this approach where multiple transmit synthesis outputs,either synthesized channel data sets, or synthesized area/volume formedoutputs, are computed and are input to the Motion Processor block, whichcombines the multiple inputs in an adaptive manner, i.e., responsive tosignal level, etc., to form a motion compensated output. Otherrearrangements of the Transmit Synthesis, Area/Volume Formation, andMotion Processor blocks are possible as well as combining variousaspects of FIG. 11A with FIGS. 11B and 11C.

Compatibility with Other Techniques

The transmit synthesis technique described is compatible with many othertechniques used in ultrasound imaging.

(1) The transmit synthesis technique described by the present inventionmay be used in conjunction with many of the other techniques which seekto provide a dynamic transmit focus. For example, the technique ofcompositing multiple transmit foci in time along the same direction,however at different focal ranges, can be used with the transmitsynthesis technique by spatially compressing the actual transmit beamaround each transmit focal range using the receive signals acquired fromeach sequential transmit firing. In fact, if computational resources arelimited, i.e., number of transmit synthesis filters, this provides a wayin which to extend the transmit depth of field around each sequentialtransmit focal range using the spatial compression aspect of transmitsynthesis of the present invention, and reduce the number of sequentialfocal ranges required to cover a given display depth. Transmit synthesisof the present invention also works in conjunction with other types oftransmit beams, not simply those of a single fixed focus location. Theactual transmit beams used may be of any type, namely transmit beamsformed using composite transmit delay/phase/apodization profilescomputed for different focal ranges, transmit beams formed by the linearsuperposition of transmit waveforms each of which are computed usingdifferent delay/phase/apodization profiles, different transmit waveformsfor each element, etc.

(2) The transmit synthesis technique described by the present inventionmay be used in conjunction with transmit temporal coding techniques suchas linear FM chirp waveforms, frequency/phase modulated waveforms,binary type codes such as Barker, Golay, etc. In fact, the actualdecoding may be included within the transmit synthesis filters directlyand depending upon the actual transmit synthesis-area/volume formationarchitecture, the decoding can operate on individual channel data whichmay improve the performance of the coding technique. Transmit synthesisof the present invention can be thought of as a type of spatial decodingwhere transmit diffraction away from the transmit focus has left thetransmit beam incompletely decoded in a spatial sense. Thus, theaddition of temporal decoding to the transmit synthesis filters createsa multi-dimensional decoding filter topology. In addition, orthogonal,or low cross-correlation codes may be employed for decoupling transmitbeams fired simultaneously, thus improving the echo acquisition rateeven further. The decoding may also be incorporated into the transmitsynthesis filter topology, or kept as a separate processing step.

(3) The transmit synthesis technique described by the present inventionmay be used in conjunction with various forms of linear and non-linearimaging, with or without the addition of contrast agents. Transmitsynthesis of the present invention can have distinct advantages incontrast agent imaging since in many cases, minimal contrast agentbubble destruction is desired. First, using the present invention oftransmit synthesis, a reduced number of transmit firings are needed toadequately sample the image. Second, the temporal and spatialcharacteristics of the transmit beam used for acquisition can besuitably optimized for minimal bubble destruction. Then on the receiveside, the present invention can synthesize a transmit beam of temporaland spatial characteristics optimized for imaging, for exampleincreasing SNR in regions where the SNR resulting from the actualtransmit beam was deficient, in order to minimize bubble destruction,etc.

(4) The transmit synthesis technique described by the present inventionmay be used in conjunction with aberration correction techniques whichseek to correct the element-to-element delay/phase/apodization errorsacross the transducer array, on either transmit, receive, or both,associated with propagating through the layers of the body which may beof variable sound speed. Transmit synthesis of the present invention mayimprove the performance of the aberration correction techniques whichmake use of temporal cross correlations between receive channels bysynthesizing a more focused transmit beam throughout the region ofinterest, which if performed prior to area/volume formation, will createa higher SNR, more correlated receive channel dataset from which toperform cross correlations. In addition, the element-to-elementdelay/phase/apodization corrections provided by the aberrationcorrection algorithm can then be used to adapt the transmit synthesisfilter values in order to improve the synthesized transmit beam'scharacteristics. This process can be repeated in an iterative manner ifdesired, improving both the transmit beam's characteristics, andelement-to-element correlation, on each iteration.

(5) The transmit synthesis technique described by the present inventionmay be used in conjunction with sound speed correction techniques (seeconcurrently filed, commonly assigned U.S. patent application Ser. No.11/492,557, entitled ABERRATION CORRECTION USING CHANNEL DATA INULTRASOUND IMAGING SYSTEM, which is incorporated herein by reference inits entirety) which seek to correct for the average, or bulk, soundspeed of the body, i.e., the body sound speed may be lower or higherthan the industry assumed standard of 1540 m/s. These techniques maygenerate multiple images assuming different sound speeds, where thegeneration of transmit beams, receive beams, or both, employ the assumedsound speeds. By quantitatively selecting the image for best focusingusing a suitable metric, an estimate of the body sound speed isobtained. The transmit synthesis filter values can then be modified inan adaptive manner making use of this estimated body sound speed, inorder to provide a synthesized transmit beam of enhancedcharacteristics, i.e., better focusing properties, etc.

(6) The transmit synthesis technique described by the present inventionmay be used in conjunction with various forms of compounding, namelyfrequency compounding, spatial compounding, etc.

For all these techniques, as long as the pre-detected, i.e., coherent,temporal and spatial characteristics of the transmit beam(s) sweptthrough the region of interest are sufficiently predictable, eitherthrough theory, measurement, or any combination thereof, then they maywork in conjunction with transmit synthesis of the present invention.

The figures and examples provide in this disclosure are for illustrativepurposes only and are not limited to the examples shown. While lineartransmit synthesis filters and linear operations in area/volumeformation were used in the examples, they are not limited to beinglinear in nature. The transmit synthesis filters and area/volumeformation may make use of non-linear, or adaptive elements in order toenhance performance under various conditions.

It is to be understood that the above description is intended to beillustrative and not restrictive. Many embodiments will be apparent tothose of skill in the art upon reviewing the above description. Thescope of the invention should, therefore, be determined not withreference to the above description, but instead should be determinedwith reference to the appended claims along with their full scope ofequivalents.

1. An ultrasound imaging method for achieving transmit and receivefocusing at every echo location within a region of interest, the methodcomprising: providing a probe that includes one or more transducerelements for transmitting and receiving ultrasound waves; generating asequence of spatially distinct transmit beams which differ in one ormore of origin and angle; determining a transmit beam spacing based upona combination of actual and desired transmit beam characteristics, thetransmit beam spacing being determined without consideration ofround-trip transmit-receive beam sampling requirements; storing coherentreceive echo data, from two or more transmit beams of the spatiallydistinct transmit beams; combining coherent receive echo data from atleast two or more transmit beams to achieve a spatially invariantsynthesized transmit focus at each echo location; and combining coherentreceive echo data from each transmit firing to achieve dynamic receivefocusing at each echo location.
 2. The method of claim 1, wherein theprobe is a 1-D or 2-D array with varying degrees of elevationbeamforming control, including aperture, delay, and phase, or a general2-D scanning array, or a sparse 1-D or 2-D array.
 3. The method of claim1, wherein the spatially distinct transmit beams are generatedelectronically, mechanically, or any combination thereof, to scan a 2-Dplane or 3-D volume.
 4. The method of claim 1, wherein coherent receiveecho data is receive channel data from different transducer elements. 5.The method of claim 1, wherein combining coherent receive echo data fromeach transmit firing is performed prior to combining coherent receiveecho data from at least two or more transmit beams.
 6. The method ofclaim 5, wherein combining coherent receive echo data from each transmitfiring is performed prior to storing coherent receive echo data.
 7. Themethod of claim 1, wherein combining coherent receive echo data fromeach transmit firing is performed subsequent to combining coherentreceive echo data from at least two or more transmit beams.
 8. Themethod of claim 1, wherein combining coherent receive echo data fromeach transmit firing and combining coherent receive echo data from atleast two or more transmit beams are performed simultaneously.
 9. Themethod of claim 1, wherein combining coherent receive echo data fromeach transmit firing and combining coherent receive echo data from atleast two or more transmit beams are performed at arbitrary echolocations or along multiple ray-like paths.
 10. The method of claim 1,wherein one or more non-linear components of the transmitted ultrasoundwaves are included in the steps of storing coherent receive echo data,combining coherent receive echo data from each transmit firing andcombining coherent receive echo data from at least two or more transmitbeams.
 11. The method of claim 1, further comprising forming an imagewith an imaging mode that includes one or more of B, color Doppler,power Doppler, M, spectral pulsed-wave Doppler, with or without usingcontrast agents.
 12. The method of claim 11, wherein velocity imagingmodes including color velocity and spectral pulsed-wave Doppler areachieved using motion compensation in combining coherent receive echodata from at least two or more transmit beams.
 13. The method of claim11, wherein the spectral pulsed-wave Doppler mode involves processingone or more sample volumes along an arbitrary path in 2-D or 3-D space.14. The method of claim 1, wherein combining coherent receive echo datafrom at least two or more transmit beams includes combining using one ormore of delay, phase, amplitude, and convolution.
 15. The method ofclaim 1, wherein combining coherent receive echo data from at least twoor more transmit beams is responsive to echo location, as well as to thespatial and temporal characteristics of the actual and/or desiredtransmit beam characteristics, which includes one or more non-linearcomponents of the transmitted ultrasound waves.
 16. The method of claim15, wherein combining coherent receive echo data from at least two ormore transmit beams includes combining using one or more of delay,phase, amplitude, and convolution.
 17. The method of claim 1, whereinthe spatially distinct transmit beams differ in one or more of delay,phase, apodization, amplitude, frequency, or coding.
 18. The method ofclaim 1, wherein the spatially distinct transmit beams have a singlefocus at a predetermined range, with an F-number ranging from 0.5 to 10.19. The method of claim 1, wherein two or more of the spatially distincttransmit beams are fired simultaneously, or with a time gap less than around-trip propagation time, for an even faster echo acquisition.
 20. Anultrasound imaging system for achieving transmit and receive focusing atevery echo location within a region of interest, the system comprising:a probe that includes one or more transducer elements for transmittingand receiving ultrasound waves; means for generating a sequence ofspatially distinct transmit beams which differ in one or more of originand angle; means for determining a transmit beam spacing based upon acombination of actual and desired transmit beam characteristics, thetransmit beam spacing being determined without consideration ofround-trip transmit-receive beam sampling requirements; means forstoring coherent receive echo data, from two or more transmit beams ofthe spatially distinct transmit beams; means for combining coherentreceive echo data from at least two or more transmit beams to achieve aspatially invariant synthesized transmit focus at each echo location;and means for combining coherent receive echo data from each transmitfiring to achieve dynamic receive focusing at each echo location.
 21. Anultrasound imaging system for achieving transmit and receive focusing atevery echo location within a region of interest, the system comprising:a probe that includes one or more transducer elements for transmittingand receiving ultrasound waves; and a processor configured to: generatea sequence of spatially distinct transmit beams which differ in one ormore of origin and angle; determine a transmit beam spacing based upon acombination of actual and desired transmit beam characteristics, thetransmit beam spacing based upon determined without consideration ofround-trip transmit-receive beam sampling requirements; store coherentreceive echo data, from two or more transmit beams of the spatiallydistinct transmit beams; combine coherent receive echo data from atleast two or more transmit beams to achieve a spatially invariantsynthesized transmit focus at each echo location; and combine coherentreceive echo data from each transmit firing to achieve dynamic receivefocusing at each echo location.